In CGEL, the ‘?’ annotation means that the sentence is of questionable grammaticality. That is, CGEL is saying that sentence [ii] with that is of questionable grammaticality, whereas PEU is saying that the quoted sentence with that is grammatical.
The problem stems from the various points of view held in these sources. First, usage and grammar are not the same thing, even if while doing grammar an author will also consider usage.
(Grammarist) In linguistics, the term usage relates to the habits of language use among a
language’s native speakers, particularly with regard to the meanings of words and phrases. Grammar relates to the functions of words, the construction of sentences, and how words combine to make sentences. Usage evolves continually, often rapidly, and is subjective in many ways. Grammar is more rule-based and objective and tends to evolve more slowly. The two subjects are often lumped together, usually under the term grammar, but for people who study and teach about them, they warrant very different approaches.
PEU is concerned essentially with usage, as the title suggests, whereas CGEL has a strictly grammatical point of view. There is yet another criterion a grammarian may refer to, and this one is user acceptability. The fact that native users are not sure about acceptability is symbolized by "?" in CoGEL; there are therefore two important types of usage: widely accepted usage and usage that is still open to question by a significant proportion of the users (let's call it "unsure usage" for the sake of referring to it). It is such usage that might be characterized by such comments as the one made above by user Kate Bunting (I would say "The place [that] we stayed in or at. Otherwise it sounds odd to me, since place isn't the direct object of stay.). Here is what the grammarians who wrote CoGEL have to say about the use of relatives in the present context (note that the symbol "?" concerns acceptability, more exactly, "unsure acceptability").
With a general antecedent, such as place, we may find the following patterns, which however are not acceptable to all speakers:
• (?)That's the place she stays when she's in London. 
• (?)That's the place she works. 
• (?)That's the place she studies. 
However, a final at […] would be fully acceptable, at least in familiar
• That's the place she stays at when she's in London. [1a]
• That's the place she works at. [2a]
• That's the place she studies at. [3a]
With a generalized antecedent such as way, expressing direction, we usually have zero rather than that:
- Was that the way she went? ['Was that where she went?']
So, PEU is not saying that the sentence with "that" (only "that") is grammatical but instead that it is fully acceptable and they stand in contradiction with CoGEL since the grammarians of that latter grammar assert that this is not yet a fully accepted construction. CGEL is not to be questioned since they seem to place themselves on the level of strict grammar and that the point that they would most certainly make (not a direct object) is correct.
Are both PEU and CGEL right about the use of that in their sentences? If so, how can we distinguish between the two sentences? If not, is PEU or CGEL wrong about the use of that in their sentences?
Were they using one and the same criterion for determining idiomaticity, one of the two only can be wrong, but one of them must be. If it is PEU, this is so on the ground of the possible evidence that could be read in CGEL and CoGEL, the former only contesting (rightly) grammaticality and therefore denying the possible reality of a usage tending towards acceptability and the second recognizing at most an unsure usage (and thereby acknowledging implicitly a breach of grammar). If it is CGEL, this is so on the ground of their failure to evidence a well accepted usage that makes of such phrases as "place they go" true idioms of the language. As it is likely that there is no such common criterion neither is right or wrong; what remains certain is that those forms have gained in acceptability to an important extent, as CoGEL acknowledges.
A distinction as to register is a matter of personal choice for the time being, but I feel that the impression of oddness is bound to surface one day or other in the mind of the reader that has become sufficiently familiar with the language.