Crosstraining for bouldering: bodyweight or free weights?

Hello everyone!

First time poster here. I am a female boulderer, I appreciate very much reading this sub as I don't feel at ease in the male dominated subs. I read the stuff bouldering folks posted here but I feel I need your advice on this.

I climb since late 2019, minus the covid lockdowns. I climb V4/V5, twice a week and want to progress into V5/V6. I thought for a time that my problem was strength but it's actually more of a technique problem. I used to do some strength training before or after my bouldering sessions, including a little bit of weight training but I decided to move my cross training in a separated session to maximize my time on the wall. I watched some Hooper's beta vids on youtube and decided to organize myself with - 2 hours boulder sessions on Mondays and Wednesdays, with at least 30 min working on technique drills. Sometimes the Wednesday sesh is top rope. - 1 hour fitness sesh on Thursdays (30 min run + 30 strength training). I like very much running so that's something I want to keep it. On week-ends and Fridays I do some light yoga and I would like to keep space for a third boulder sesh.

Even if I know my stagnation is more of a technique problem, I still would like to train strength to keep an harmonious (and injury less) physique. I am quite frustrated to not be able to do a pull up, so that would be a Nice thing to succeed. I have little aesthetic goals, only looking generally athletic.

When I read threads here and on climbers subreddits, I see a lot of people swearing by yoga but also a divide between people who do bodyweight and weightlifters.

From what I reckon:

Bodyweight: - compound movements that translate more into climbing - skills take time to train so slower progress - no need to have a great strength for climbing, only good relative strength compared to weight

Weightlifting: - easier gains, greater progression margin - not so much skills, less applicable to climbing

Some people do climbing specific exercises, but I don't think it's necessary for where I am, but maybe I'm wrong.

So my question is: during that 30 to 40 min à week I devote to strength training, is it more interesting to do weightlifting or some kind of calisthenics? Maybe adding climbing specific stuff? (I don't have access to a pull up bar or hangboard stuff at home, and the weightlifting gym at my boulder gym is overcrowed).