I think "swapping" is a strong word. It's the VIN from the donor car, so I think it's a defendable position to use it. If Montana registration is currently working, that's great. I'm definitely not claiming this is a better option, just another idea to consider.
Since Covid, I've not encountered any police officer who had much of any interest in looking for extra work to do. Four months ago I got pulled over for a burned out license plate bulb at 2am. I thought it was some kind of profiling, but I did check and he was right, both bulbs were out. I accidently gave him the registration for the wrong car(it's a long and unrelated story as to why that registration was in the glovebox). He wrote up a warning with wrong tag, vehicle type(pickup vs sedan), make, model, etc. I don't think he cared at all.