The Super Cat Tales games use Adobe AIR? That's pretty cool!
I started playing through the first two a while ago, and they're a lot of fun. I'll check out the new one too!
With Desktop AIR, I use Zeh Fernando's KeyActionBinder class to add controller support. It may be a little bit out of date now, but it worked well for me for all of the common controller types, and was fairly easy to use.
I'm not sure how well it will work on mobile - I tried running my Desktop game Bullet Heaven 2 on an Android device with an Xbox One controller, and it kind of worked - but I think the controller buttons were mapping to keyboard keys instead, or something weird like that. So I think it can work with some tweaking.
If you decide to try it, let me know if you have any issues. I had an issue with the json file not being included correctly, so I hacked the code a bit to take it as a parameter. I also had to change a few references to "Windows 7" to just "windows" in the json file to make it work on my PC.