Total Control 3 odd behavior

[TheRedFox]

So, I have a total control 3 (SAT controller to DC adapter). The button mapping seems to make sense, with ABXY the same, and RL mapped to CZ as well as RL, but it won't let me use the saturn controller's RL buttons in one specific situation.
In the game Gunbird 2, R / C is default mapped to autofire (it supports the arcade stick, so that's why it has C listed in the controls). In game, then, the C will do its autofire, but R will do nothing.
That maybe would make sense, except that RL still works in the pause menu, and art gallery for what it would normally do.
On the other hand, in Sturmwind, R and L behave as normal (which works well for that game).
I did figure out a button mapping that works fine (with B as autofire, A still as charge shot, Y close attack, and C bomb), but I think it's a little easier to do this if I can just lean on the R button the whole time.
one more thing that I suspected, and sort of confirmed (via sega retro wiki article) is that the SAT controller is identified as an arcade stick by the DC, although choosing different arcade stick button layouts on Sturmwind didn't change a thing (it just gives me the default DC controller layout whichever I choose, which is fine, b/c the shoulder buttons work)

Any thoughts on why this might be, or how I could trick it back into R for autofire?