I've been bouldering for 10 years, messing around with some gymnastics elements for a few years, and feel I can comment on this with some authority. Just about any bodyweight routine will help your bouldering A LOT. The unique stuff it does for the stabilizers and core carry over very well to climbing. To wit: when I worked at a climbing gym in town you could always tell who were the experienced yoga/gymnast types. Where newbies often struggled with basics, these people (still newbies, mind you) were only limited by their pain tolerance and understanding of technique. They would often cruise 5.10c/d toprope routes on their first trip to the gym. That won't mean much to a non-climber.. but it's really impressive to watch in context.
If you plan to train both, you do need to be careful though.. the demands on your shoulders, especially with how short you are, are going to be significant. At your height you will need to go to full extension to do moves on boulder problems that someone only a few inches taller will cruise easily, and that part of shoulder ROM is where most climbers get hurt. Adding gymnastics training, potential scar tissue and an imbalance between prime movers and stabilizers isn't going to help. My point? Make SURE you take your time to warm up and recover.
Also, be sure you work opposing muscles. Climbers often decide, after years of climbing, that they should start doing pushups to balance everything out. They then get injured. If you develop the pressing/pushing muscles from the outset you'll have healthier shoulders (and posture) in the long run. Plus the press elements of gymnastics are a lot of fun, and damn impressive, especially when it's a grown woman doing them.
BtGB is an awesome place to start, and should serve you well. Just figure out where your weaknesses are and focus on that. I'd say the whole gymnastics program, plus a healthy bouldering/sport climbing obsession, are probably too much in combination. That would be a TON of abuse on your upper body.
Which tendons are you worried about specifically? Finger flexors shouldn't be an issue with the gymnastics, but certainly could be with climbing. Muscle grows a lot faster than tendon, and while we are wired up to protect the tendons if they get overloaded, it DOES happen. I've had issues with the distal biceps tendon off-and-on for years. Through trial and error I've learned that when my diet is really clean (ie: anti-inflammatory) and I warm up adequately, I don't have problems. It goes back to the bit about warming up and taking time for recovery. This will be very important for what it sounds like you want to do.
I wish you well with it.. I'm always excited when I hear about someone going through that initial phase of getting stoked on climbing.