I went ahead and cut the pieces down to give it a try since I have plenty of scrap to play around with. I cut the plywood to 19x10" and the 2x2 to 19" and screwed the plywood to it.
Turns out it wouldn't fit behind the Brake Buddy with the 2x2 against the seat track, just a bit too close. BUT I flipped the whole thing over so the edge of the plywood was on the floor in front of the seat track and the 2x2 was up against the handle where the seat is supposed to be pressing and it actually fit quite well.
I fired up the unit and started the auto setup sequence so it would pump the brakes and it didn't twist and rock like before so I think we have a winner!
The previous Brake Buddy design had the rear handle stick out past the back of the unit with an up/down adjustable position and I think that was actually the better design. I think they came up with this one with the assumption that the unit itself would fit underneath the seat a bit so the handle would always press into the seat front but that doesn't work here.
Now that I have this working I have an idea to replicate that old design using this 2x2 or a 2x3 as the cross bar part of that handle at the height of the 2x2 in my picture. That would let it reach the seat cushion and press on it instead of the bulge on the back of the main unit hitting the plastic trim piece first. It'd take up less space than the chunk of plywood too.