Swap Right and Middle mouse buttons

I have a three button mouse, but, for some reason, the middle mouse button generates a right click and the right button generates a middle click. I want to swap the two buttons.

To make this discussion easier, let's call the three physical buttons 1, 2, and 3, based on their physical positions from left to right, and call the signals left, middle, and right.

What I want to do is make button 3, which normally generates a middle click, generate a right click. This seems easy enough with BTT.

But I also want button 2, which normally generates a right click, to generate a middle click. But when I try to set this up in BTT, it says "needs modifier key" and apparently won't let me proceed to remap it without one. Is there any way around this?

did you ever get an answer to this? i'm in a similar boat and can't change my physical right mouse button to be something else without a modifier key. i don't want to hold a key on the keyboard to click my mouse. that's ridiculous. if the dev is scared someone is going to map themselves out of left click, then make sure you have it assigned to another button before letting them save.