On the other hand, if you charge the developer (and not the plateform) to check certificate validity, it means that you have to develop a different codebase for all plateform you are targeting, because each plateform store trusted root certificate in a different manner with different APIs, and also have different types representing a X509 Certificate.

That's what cross-platform abstraction libraries are for. Both Java and Qt provide a key store library that can load from either the OS root store or a custom one. If your chosen app platform doesn't, OK, then you'll have to make or find one yourself. Perhaps contribute it upstream or make it a library. But that's not a limitation of BIP70.

Just as a reminder, there is no obligation to use the OS root store. You can (and quite possibly should) take a snapshot of the Mozilla/Apple/MSFT etc stores and load it in your app. We do this in bitcoinj by default to avoid cases where BIP70 requests work on some platforms and not others, although the developer can easily override this and use the OS root store instead.

Of all possible solutions, using a third party service to convert things to JSON is one of the least obvious and highest effort. I don't know anyone else who arrived at such a conclusion and respectfully disagree that this is a problem with the design choices in BIP70. It sounds like a bizarre hack around lack of features in whatever runtime you're using.