I responded to their complaint of missing titles, asking how best to solve it, which implies that I would need to know how they actually produce the paper, e.g. do they work from the PDF I generated, an automatically generated (from my LaTeX files) PDF (which would have the incompatibility problems mentioned above), do they convert the LaTeX files to some other format and process that? I included in my reply a .bbl file in the wrong format but with the titles and asked them if they needed anything else. I didn't receive an answer, but received the proofs a few days later, with the notice that they would be the only proofs I would see before publication (and presumably there would be none afterwards).
Looking at the proofs, thankfully the main text came through unscathed. References in the text were changed to use numerical citations only, rather than the natbib style I had used (which has just numbers for the equivalent of "(Jones 2023)" but adds the name before the number for the equivalent of "Smith (1991)", e.g. "[1]" and "Smith [2]". Since it is unclear how that was done, I decided to leave things as they had been given to me, and keep in mind to use strict numerical citations in the future. But I might ask them to change their BibTeX style file to include titles or make one of my own; while the latter would be OK for me, whether it would help with the missing titles depends on whether they would use my BibTeX style file, assuming that they use LaTeX/BibTeX at all. (However it was done, it was obviously not done automatically, so any attempt to change things—which might not be compatible with the journal style anyway—would have taken a significant amount of time. Not worth the effort; life is short.)
In any case, except for the lack of authors' names in combination with numerical citations in the text, apart from minor matters of formatting my author's accepted manuscript and the final version are identical, except that mine has no titles in the references and some of the boilerplate obligatory statements at the end are a bit different (it wasn't clear which are required), and that the standard short statements are after rather than before the appendix. It doesn't seem to be worth the trouble to produce my own version with references, as the official version is publicly available. In the official version, reference numbers are links, so one can thus see the author even when not mentioned in the main text.
If someone knows how to use LaTeX and BibTeX to produce a manuscript for Foundations of Physics which will result in proofs in the format the journal wants, then I would be interested in hearing about it. Using the supplied macros (but without natbib) would presumably leave out the titles. Step 1 would be for their own macros to include titles if that is desired. Step 2 would be to make their macros work properly with natbib and allow a natbib-style submission (but with numerical citations) which would go through the production process unchanged.
I hadn't tried to solve this problem before submitting the manuscript, as it would have been wasted effort (at least for the time being) had the paper (my first for Foundations of Physics) not been accepted. It still isn't clear how much work would be needed nor whether such work would actually result in a proof in the correct format.
As long as it is possible to publish in Foundations of Physics with no charge to me as an author and my personal minimum regarding open access applies (I am allowed to put the author's accepted manuscript on my own web server with no embargo period), then I would like to publish there again in the future. However, most of my work is not on the foundations of physics, so I don't know when, if ever, I will publish there again, even assuming that the current conditions apply.