I'm kind of the opposite. I went full cave and really don't see a need for trimix at the moment for the type of diving I do.
I was going to suggest the same thing for just doing cavern.
Cavern is where you learn all the skills needed to dive in caves, or all those skills you mentioned wanting to learn to be a better diver.
Cave is just learning how not to die in a cave using said skills you've already obtained.
Why not do the cavern portion of the cave series of classes then go for the trimix.
This way you will build up a solid foundation of skills with some task loading expectations/experience and some problem management skills.
Also you would get a sense of a cave(ern) environment without committing to far along the path of your goals
I was going to suggest the same thing for just doing cavern.
Cavern is where you learn all the skills needed to dive in caves, or all those skills you mentioned wanting to learn to be a better diver.
Cave is just learning how not to die in a cave using said skills you've already obtained.