I did mention in my question that there is one piece of evidence that is red is a constituent: it can be coordinated with something that is definitely a VP, as in
 The hat is red and weighs 151 g.
I also said that this is weak evidence because it is known that one can sometimes coordinate non-constituents.
However, I now think that if one supplements  with additional considerations, as I will do below, this will turn it into a much stronger argument---possibly strong enough to be acceptable as the answer to my question.
Please tell me what you think by upvoting (or downvoting) and by commenting. (But if you do downvote, please explain why.) Whether I accept this answer or not will depend on the feedback I receive (unless, of course, a clearly better answer appears in the meantime).
The required additional considerations are outlined in CGEL (pp. 1348-1350): 'Coordination as evidence for a VP constituent'. Basically, we will ask 'what else could is red in  be but a constituent?', consider all known alternatives in turn, and conclude that the analysis where is red is a constituent is better than all the alternatives.
What CGEL says is that we have the following principle:
 In general, if a sequence X can be coordinated with a sequence Y
to form a coordination X and Y, then X and Y are constituents,
Coordination clearly does not provide a simple and absolute criterion
for constituent structure: the qualification 'in general' in  is
indispensable. It nevertheless remains a useful criterion: if a
sequence X can be coordinated, then the simplest account will be one
where it is a constituent entering into basic coordination, and we
will adopt some other, more complex, analysis only if there are
independent reasons for doing so.
Thus we will say that X is a constituent if that is the simplest analysis that accounts for all facts---and that is usually the most we can ask for in any analysis, linguistic or otherwise, for there is no upper bound on how complex an analysis one can invent.
CGEL's discussion uses the sentence Sue found the key as its test case. I will reproduce what they say about it, and then, at each stage, discuss how what was said relates to The hat is red.
How does coordination support the analysis of a clause like Sue found
the key into two immediate constituents, as in [36i], rather than
three, as in [ii]?
 i Sue | found the key.
+ VP: the VP analysis]
ii Sue | found |
the key [NP + V + NP: the 'no VP' analysis]
Principle  supports the VP analysis
The sequence found the key can be readily coordinated:
 Sue found the key and unlocked the door.
Only analysis [36i] is therefore consistent with principle ; other
things being equal, it is to be preferred over [36ii] because it
allows us to subsume  under basic coordination, so that it
requires no special treatment. Principle , however, is qualified,
not absolute, so we need to consider the matter further.
So far, word for word, the discussion applies to The hat is red: does the sentence have two immediate constituents (The hat | is red) or three (The hat | is | red)?
The additional considerations
VP-coordination vs clause-coordination with ellipsis of the subject
An alternative treatment of the coordination here, one that is
consistent with [36ii], is to say that the coordinates are clauses,
the second having ellipsis of the subject. The first coordinate will
then be not found the key but Sue found the key, and the second
will be ̲ ̲ unlocked the door. This
accounts for the equivalence between  and Sue found the key and
she unlocked the door. But we saw in §1.3.1 that there are many
cases where no such equivalence obtains, as in:
 i No one treats me like that and gets away with it.
ii No one treats me like that
and no one gets away with it. [clause-coordination]
An elliptical clause analysis doesn't provide a satisfactory account
of coordination like [38i].
In our case, the challenge to explain why The hat is red and weighs 151 g must be an instance where we coordinate the VP is red with the VP weighs 151 g (so-called 'VP-coordination'). Why couldn't it instead be an instance of 'clause coordination', where we have an ellipted subject in the second coordinate? In other words, why can't we say that The hat is red and weighs 151 g is completely equivalent to The hat is red and [it]/[the hat] weighs 151 g? It seems that either reading would work equally well.
CGEL points out that in some settings, the two readings do not work equally well. So consider No hat is red and weighs 2 lbs, and note that it does not mean the same thing as No hat is red and no hat weighs 2 lbs . The former only says that there are no hats that are both red and weigh 2 lbs, whereas the latter says that there are no red hats, whatever their weight, and also that there are no hats that weigh 2 lbs, whatever their color. We conclude that The hat is red and weighs 151 g is better analyzed as a VP-coordination than as a clausal coordination.
Basic vs right nonce-constituent coordination
Another alternative consistent with [36ii] would be to say that the
underlined sequences in  are merely nonce-constituents,
constituents in the coordination but not elsewhere. This is to treat
 like our earlier example:
 I gave $̲1̲0̲ ̲t̲o̲
̲K̲i̲m̲ and $̲5̲
̲t̲o̲ ̲P̲a̲t̲. (=[i6ii])
There are, however, two important differences between  and .
In the first place, the nonce-constituents in  have to be parallel
in structure (as noted in §4.3), whereas those in  do not -
compare, for example, Sue found the key and departed. Secondly, the
reason why $10 to Kim is not a normal constituent is that there is
no direct syntactic relation between the parts, $10 and to Kim:
they are, rather, separately dependents (complements) of give.
Found the key in  is quite different: here there is a syntactic relation between the parts, found being head and the key
dependent. The nonce-constituent analysis is a more complex type of
construction than basic coordination, applying under restricted
conditions (the requirement of parallelism) and justified by strong
independent arguments against recognising the coordinates as normal
constituents: in the case of  we have no reason to prefer the more
complex analysis to the one that follows the general principle given
In our case, we also lack any requirement of parallelism (e.g. The hat is red and reeks), and we also have that there is a syntactic relation between is and red, namely is is the copula and red a predicative complement of it. Thus, like CGEL, we have no reason to prefer the more complex nonce-constituent analysis to the one that follows the general principle given in .
Basic vs delayed right constituent coordination
The relevance of  to constituent structure might be challenged on
the grounds that it is also possible for coordination to group found
 M̲a̲x̲ ̲l̲o̲s̲t̲
̲f̲o̲u̲n̲d̲ the key.
If coordination can group found with either the key or Sue, the
argument would go, then it can't provide evidence for a constituent
grouping of found with just one of them, the key. But such an
argument fails to recognise that there is a major difference between
the coordination of  and that of . The latter represents a
much less usual type of coordination than , and this instance of
it is indeed of somewhat marginal acceptability because of the low
weight of the key; acceptability is increased by expanding to the
key to the safe but greatly diminished by reducing to it. Example
 would characteristically have special prosody, with a clear break
before the key. [These properties do not hold for all cases of
delayed right constituent coordination, but even when they do not
there will be independent evidence for treating the coordination as
non-basic. Take, for example, [24iii], He was accused but found not
guilty of stalking a woman for seven years: it is evident that in the
non-coordinative He was found not guilty of stalking a woman for
seven years the of phrase is a complement of guilty, not find
not guilty, because it regularly occurs with guilty quite
independently of the presence of find, as in He was/seemed guilty
of treason.] But , by contrast, has no such limitations or
special prosody, and can be taken to represent the most elementary
type of coordination: as such, it does provide valid evidence in
support of the VP analysis.
Again, what CGEL says of its example, we may say of ours. Consider the following instances of 'delayed right constituent coordination' (CGEL's preferred term for what everyone else calls right node raising):
 a. The hat is but the scarf is not red.
b. The scarf only seems but the hat is red.
As in CGEL's example , [2a] and [2b] are of marginal acceptability, which increases if the 'weight' of the 'extracted element' is increased:
 a. The hat is but the scarf is not a family heirloom left to me by my grandpa.
b. The scarf only seems but the hat is well over 100 years old.
Also, when [2a] and [2b] are pronounced, there will characteristically be a prosodic break before red.
In contrast, there are no special limitations on the 'weight' of what can be coordinated with is red at a given level of acceptibility: on one extreme we may coordinate with a single word, The hat is red and reeks, on the other with a much longer phrase, The hat is red and makes me think of my childhood summers which I spent with my grandpa, and both are equally acceptable. Also, there is no special prosodic break in any of these.
We have contrasted the analysis of
 The hat is red and weighs 151 g.
(and similar sentences) in which is red is a constituent to three possible alternatives: 1. one where  coordinates clauses and the second clause has subject ellipsis; 2. one where is red and weighs 151 g are nonce constituents; and 3. one where is is argued to be just as easily grouped with the hat as it is with red. We found that in all three cases, the analysis where is red is a constituent is simpler, more plausible, or both. This gives us solid evidence that is red is therefore indeed a constituent.