After 7 days of the bounty period not attracting any answers, I'll answer this question to the best of my abilities.
General considerations
Geometry optimization with one method, and properties with another
This is fairly ubiquitous in matter modeling. High-accuracy methods such as FCI, CCSDTQ, and even double hybrid functionals are almost never used for geometry optimization because the computational cost would not be worth it, yet we still have those methods implemented and frequently used, meaning that they are frequently used on geometries that either come from experiments or from other computational methods.
Geometry optimization with one method, and dielectric functions with another
The link you provided about ionic and electronic contributions to the dielectric function, doesn't say anything about needing to use the same functional as the one that was used for the geometry optimization. It provides some VASP output saying that meta-GGA functionals can't be used with DFPT (implying that they can be used when doing finite differences):
#The ionic dielectric function can be calculated in two ways:################
#1# DFPT (faster), but does not allow for METAGGA use. ######################
IBRION = 8; LEPSILON=.TRUE.
#2# Finite differences (slower). ############################################
but if it was so important for the functional used for the dielectric function to be the same as the one used for geometry optimization, then it's peculiar that they never mentioned anything about that in the tutorial. The words "forces" and "force" never show up in that article either.
The article also cites two papers. This one never mentions "force" or "forces" and this one barely does, but a deeper look at those papers would be needed to see if they use (or recommend) to use the same functional for the dielectric function calculation as for geometry optimization.
Pragmatic considerations
If it was pragmatic to do the geometry optimization with the hybrid functional, then you would (pragmatically) do it. So assuming you are limited to the cost of PBE for the geometry optimization, your choice is either to do the dielectric function calculations with PBE (same functional) or a different functional. When using the PBE geometry, does the dielectric result depend a lot on the functional? If no, then you can report results with all functionals that you used and say how small the spread of results was.
You can also do the geometry optimization with other functionals that have similar cost to using PBE. When using the hybrid functional for the dielectric function, for various geometries that were optimized with cheaper functionals, does it depend a lot on the geometry? If no, then you can report results with all geometries that you used and say how small the spread of results was.
What if there was strong dependence on the functional used for the dielectric function and/or the functional used for geometry optimization? Then it matters which functional you use, and from a theoretical standpoint the "best functional" for that system would usually be the one that you feature more prominently in the paper. Let's assume that PBE is the best functional you can use for the geometry optimization step, then our question becomes a matter of which result is better:
- PBE (geometry) + "best" hybrid for system (dielectric function)
- PBE (geometry) + PBE (dielectric function)
We can then return to your idea of comparing to experiments. Does PBE+"best" consistently match experiments better than PBE+PBE? Another thing you can do is pick a smaller system for which much more accurate calculations can be done, and see which functional/functional-combination recovers the accurate theory results the best. This might mitigate some of wzkchem5's concern about agreement with experiment being fortuitous.
Acceptance by others
In the same comment by wzkchem5 for which I provided a hyperlink directly above, there's a concern about acceptance by others (e.g. referees). As a researcher that finished your MPhil a few months ago (based on the website in your Stack Exchange profile), there would almost always be more senior co-authors that know more about what others in the same research field will accept (or at least they'd know people that could give advice about it), but not always. People sometimes work on projects outside of their comfort zone (which can be a good thing).
If no member of the research collaboration knows whether to use the same functional for both, or to use a better functional for the dielectric function, and there's no one clearly saying that you shouldn't use different functionals on an MMSE question with a bounty that lasted for its maximum duration, and no co-authors could find any advice in published papers or by personal contacts, then you can submit the paper with (PBE + "best"), knowing that one or more referee(s) might tell you that it should have been done with (PBE+PBE) or ("best"+"best"). By this time you'll probably be done the "best"+"best" calculation since it would usually take weeks or months to get referee feedback (plus whatever time it takes to prepare the paper for submission). Often, the better the journal, the better (and more scrutinizing) the referees (you can also try to give the journal good referee suggestions), but not everything gets caught by referees.
If using (PBE+"best") is really such a big deal (and was somehow ignored in that VASP webpage that you gave us), it's unlikely that no referee will comment on it, so you can be fairly confident about acceptance at that point, and the final test will be whether or not the wider community of article readers accepts the paper. Apart from the due diligence you already did here at MMSE, and all the other things I mentioned that I'd expect from the more senior (if any) co-authors (e.g. asking experts directly), you could present the work at a conference before publishing it, but other than those things there's not much else you can do to guarantee that the paper will be accepted in this wider sense, and this is a risk that's taken whenever embarking on a project like this.
