I've heard that the LP is detected by Rock Band as a regular controller (like a vocalist), and that it even works in that capacity. My suspicion would be that Rock Band simply reads the device descriptor on whatever controller you're using in order to determine what kind of instrument it is. Therefore, it seems you should be able to intercept any USB communications and change the device descriptor from "GH Les Paul" to "RB Strat" or whatever the actual signal equivalent is and solve the problem.
The case where this would not work is if the LP and Strat have different button mappings as well, although that could easily be tested by using them to navigate the XMB.
edit: By the way, somebody commented that the Strat has two device descriptors, I'd bet that they're either PS3 controller and Strat IDs or they correspond to the guitar and the wireless receiver separately, in either case I would expect you to only have to change one of them. You could probably figure out which just by hooking both the LP and Strat up to a PC and comparing the device descriptors.