OP, I'd suggest looking at
Gymnastics Bodies and
Beast Skills, as well as
EatMoveImprove. Those three sites contain lots of information, and some tutorials, about gymnastics strength skills, which are what this board generally recommends as far as upper-body bodyweight strength training.
Your bodyweight big 3 are going to be planche, back lever, and front lever (someone correct me if I'm wrong). As you may already know, lower body and posterior chain are going to be harder to train without barbells. You may want to keep squatting for simplicity's sake. If not, Natural Leg Curls/Glute Ham Raise (NLC/GHR) are excellent for the posterior chain, and combining sprints and plyos should round out the whole program rather nicely.
Honestly, I'm not entirely sure how to program these exercises if it's a bodyweight only program, but either one of those sites, or someone else here may have better specific feedback on that. I've always used levers as assistance exercises and thrown them into a modified Starting Strength split.