yes, there are a lot pre build menu library for arduinos. They should also work with teensy/-duino.
I don't know if axoloti fw sends any message if a new patch is loaded to axoloti but you can code a routine and write your "hello, I'm a new program" command and here are the start condition and values which are send one time if a new patch is loaded to axoloti and it's starting for the first time. same on your microcontroller ehich send "hey, I'm ready to receive"…
theoretical it's working like any normal midi controller which has encoders build in.
@MrBim
your project looks cool.