The question is can that system work on the same frequency, and I'd say "yes". There's no legal framework in any country I'm aware of that would allow APRS but not other paket services.
Is that a good idea? Is it socially fine ?
It really depends. Is your secondary system going to deal nicely with the APRS that other people want to use? In that case, I'd say, go for it. APRS is technically very old and while I think it enables many a valuable application, having so much respect for it that you don't use a band because someone unspecific might potentially see interference with their antique system that has no real medium access control isn't worth it.
But that's a bit of a philosophical approach I have: I come from a communications engineering background, and I know how valuable the spectrum is that amateur radio is allowed to use. I'd much rather see the technology used in that spectrum actually being closer to state of the art, meaning that the technology uses the spectrum as efficiently and briefly as possible to maximize utility. The reason is simple: if governments decided that tomorrow, the, say 2m band was instantly assigned to machine-to-machine low power wide area networks, that would definitely enable a lot of societally beneficial applications from automated agriculture to automatic bridge maintenance to better control of small photovoltaic plants through early demand information from regional electricity consumers... We don't do all these nice things with the ham bands because we assume there is cultural and technological value in what hams do with that spectrum.
AX.25 is a bit of a spit in the face of anyone who would like to use these bands for the general good (or even just to make money), in my humble opinion, it being an utterly unsuitable protocol for wide-range wireless communications. So, if you asked me, if ask you to do something modern, efficient in that band, to offer your BBS through that less wasteful physical layer. Nobody forces you to transport data using a >50 year old standard that was already heavily criticized back in the day for lack of appropriate forward error correction and checksumming, and on a physical layer that every student of electrical engineering could tell you after their digital communications 101 lecture is very spectrally inefficient.