Then I'm glad to be of at least partial assistance with your search. Ah, I'm not entirely sure if it does, and I can't presume so due to it including the entirety of the messages, and I unfortunately can't check, as I don't have any MMS to save.

I do however hope it does, but if not, I wish you the best of luck with your search, and that others will chime in with more effective solutions.

While it isn't routine, I quite like QUIK, as it allows at least manual backup of messages to a folder you set within your phone, it can be found on F-Droid, although I'm unsure if it's within any repositories or just found within. QUIK

My apologies it isn't exactly what you're looking for.