If you run them in the correct order, it should be fine. I run both.
So when repatching (after every new collection of mods), I uncheck my bashed patch, AV, ASIS Dependency, and AV (that's the order they're in, at the end of the load order).
I run Bash (having already run BOSS, of course). I activate the patch.
I run AV, and activate it. It has the bashed patch as a master.
I activate ASIS Dependency (I didn't want it in the other two patches, but I need it now.
I run ASIS. It has the last three as a master (plus all my other mods, of course!)
So far, I haven't had issues with this. I believe ASIS says in the FAQ that this is the method for running two SkyProc patchers at once.
Takes a while, especially when I'm adding one new mod, but it's worth it for the ASIS and AV goodness!
edit: Oh, and if AV has ASIS as a master, you need to run AV when ASIS isn't loaded. That should sort things.