As I suspected, there is no .PAT file for the Roland Integra-7 supplied by PG Music. There is one for the Roland Fantom, but you don't want that.

So, what are you using as your .PAT file? Everything you complained about sounds to me like a problem involving the .PAT file. If you have a good .PAT file, can you describe how you accessed it in BIAB?

There is a .PAT file on the Patch Files support page. This one is written by Dan Callaway in 2013. Is that what you have? If so, his text describes that the Integra-7 can only have 4 of the 12 SRX module sounds loaded at a given time. He chose to load 2 through 5, but gives instructions how to get the rest from Cakewalk. The biggest problem I see with his .PAT file is that you would have no idea which of the patches were from the SRX modules because his .PAT file does not have comments in the code.

BIAB 2024 Win Audiophile. Software: Studio One 6.5 Pro, Swam horns, Acoustica-7, Notion 6; Win 11 Home. Hardware: Intel i9, 32 Gb; Roland Integra-7, Presonus Studio 192, Presonus Faderport 8, Royer 121, Adam Sub8 & Neumann 120 monitors