The closest that one prominent grammar book, CGEL, comes to addressing this question is in [14ii] below (p. 461).
While your particular case isn't exactly like any of  i-iii, I don't think it is any more different from any of them than they are from each other. In other words, I would say that your example could well be  iv. And so whatever is said below that applies to all of  i-iii, the same also applies to your case.
Constructions where nominative and accusative are in alternation
(d) Subject of (other) verbless clauses
 i a. He was morose, she full of life. b. He was morose, her full of life.
ii a. What, he a republican? b. What, him a republican?
Example [i] belongs to the gapping construction (Ch. 15, §4.2); the nominative is readily used in formal style, while the accusative seems very informal or colloquial. Example [ii] belongs to the bare predication polar echo construction (Ch. 10, §4.8.3), which is itself generally a somewhat informal construction, so the accusative pronoun is more likely; nevertheless, a nominative, as in [iia], is certainly possible. There is also the verbless counterpart of the gerund-participial adjunct discussed in (b) above:
 I knew people thought ours an unlikely alliance, I neat and quiet, he restless
Again accusative me and him could be used as subject in a more informal style. In [12-13] the clauses, though verbless, nevertheless contain a predicative element. In constructions without such an element following the subject, nominative case is less likely:
 i a. Gary took the call, not I. b. Gary took the call, not me.
ii a. A: Who ordered a taxi? B:?I. b. A: Who ordered a taxi? B: Me.
iii a. A: I'm going home. B: *I too. b. A: I'm going home. B: Me too.
Few people would use a nominative in [i], fewer still in [ii], where it would sound excessively pedantic (even more so, probably, with a negative: ?Not I), and in [iii] it can be regarded as completely unacceptable. If the accusative is felt to be too informal for the context, the construction can easily be avoided altogether: I didn't take the call, Gary did; I did; So am I.[Whatever the case, the deictic pronoun I is much more likely in constructions like [14ii-iii] than the usually anaphoric he or she.]
As far as what to say to English learners...
When it comes to the subject of a verbless clause, the accusative is strongly preferred at least in the informal style. Sometimes this preference is so strong that the accusative became mandatory even in the formal style. Note the word 'sometimes', which raises the question: when, exactly, is the nominative mandatory/allowed/forbidden in the formal style? Unfortunately, as far as I know, no one (i.e. no linguist) has been able to satisfactorily describe this, especially not in terms of some sort of general rules, no matter how complicated, and so learners shouldn't hope for them. Instead, a learner simply has to be exposed to every kind of construction and learn what is mandatory/allowed/forbidden on a case-by-case basis.
Here are some general remarks from CGEL, (p. 458, boldfaced emphasis mine):
We look first at the contrast between nominative and accusative case, where we find a considerable amount of variation and instability in the system. There are a number of constructions where the nominative is associated with formal style, the accusative being strongly preferred in informal speech and writing. Because of the tendency of older prescriptive grammar to accept only formal style as 'grammatically correct', there has been a tradition of criticising the accusative alternants, and the stigmatism attaching to such accusatives has given rise to a certain amount of hypercorrection, with nominatives being used in constructions where the traditional rules call for an accusative. Or at least this is the situation with the personal pronouns and determinatives: with interrogative and relative who the reverse situation obtains, the accusative whom being the case associated with formal style.