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.
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.