My initial thoughts are - no worries. They are lower life-forms and simply exist and reproduce. if they can survive in all water conditions as they do in all freshwater aquaria then there will not be any issues... Ofcourse I dont know if it is exactly the same bacteria that live in all conditions. It could be that there are varous strains that exist.
Sorry. Its an unknown.
There are thoughts of running the filter in an intermediate water (halfway between the 2) for a while but this wouldnt do owt.
Best bet is to stop feeding the fish in the destination tank for a couple of days to reduce waste production and then switch the filters over. Then let the filter run and watch the water parameters. Start feeding the fish sparingly and return to normal feeding if ammonia stays nil.
A gently gently approach.