I'm with Custard on this - MR MDF was made for this kind of application; with (painted) ply you'll just be making work for yourself, and you'll likely end up with a finish that isn't as good. IME MDF generally takes paint very well, wether sprayed, or applied with foam roller; if you can't get a good painted finish on MDF, you're probably doing it wrong :wink:
IMHO the 'weakness' of MDF is also being massively overstated here, for the OPs application. If you're making well-designed fitted furniture, the only time a joint should be stressed is when you move the carcass into position, as it will, by definition, never move again. Like others here, I have made (and painted!) many, many hundreds of items of fitted furniture from MDF over the years using every possible jointing method, and have not had a single failure. Not one.
I don't think pocket-holes are a great jointing method generally, but I'd agree they are 'least good' in MDF - I only ever like to use them as a convenient way of 'clamping' a joint while the glue dries. If you don't feel confident enough to drill & screw through the carcass and fill the holes, then consider another way of doing things e.g. use an end panel to hide them, use some kind of KD fittings, buy a cheap biscuit jointer etc.. etc.. FWIW the fitted furniture industry came across these issues and has largely solved them, years ago...
Whatever you decide, good luck with the project, and don't forget to post pictures!
Cheers, Pete