In case anyone else runs into this issue, here is where I landed. There doesn't seem to be a good solution for using gamepads in Crostini. There is another Crostini-like project for ChromeOS called Borealis which is intended for Steam gaming and does (obviously) support gamepads but it's still in beta and it's not on our Chromebooks.
I decided to go a different direction. I'm going to use Raspberry Pi 4 as a game console. We'll develop games on our Chromebooks (without gamepads) but play them on the Pi 4 connected to a TV with gamepads. Relevant discussions are here and here if you're interested.
Update: I ended up going with an Orange Pi 4 LTS running Android instead due to Godot 4's Vulkan or OpenGL 3.3 requirement. Works great!