I don't like the order here. You have your back (pulling) day immediately after your climbing day, which stresses the same muscles. You'd be better off to do them both on the same day or separate them by at least one day of non-pulling activity. It you choose to do them on the same day, you can make the gym workout a more abbreviated version.
As for diet, you first need to estimate your Total Daily Energy Expenditure (TDEE). Only after this will you have a starting point as to what the average number of calories should be to maintain your current weight. From there,
consistently be at a 300-500 calorie deficit. Your following post about 1700-1800 is bordering crash diet territory and it won't be sustainable in the long run. If you are exercising 6 days/week I imagine your TDEE is (or will be?) closer to 2,500 (just a guess) and you'd want to shoot for 2,000-2,200. Find out for yourself, but know that the only way you can truly know this is by collecting your own personal data (calories, activity, weight) after several weeks.
https://www.iifym.com/tdee-calculator/
Lastly, if your #1 goal is to lose body fat, you're going to want to do more cardio. Since you have a lower body injury, I'd suggest substituting one of your pulling days each week for an additional swimming session. You can even alternate your pushing and pulling each week. Something like this:
Week 1:
Sunday BBT
Monday Swim
Tuesday Rock Climbing
Wednesday CTA
Thursday Swim
Friday BBT
Week 2:
Sunday CTA
Monday Rock Climbing
Tuesday Swim
Wednesday BBT
Thursday CTA
Friday Swim
Week 3 = Repeat Week 1.
This way you are swimming 2x week, climbing 1x week, and hitting your pushing and pulling muscles with weights every 5 days for each muscle group. Your muscles will get a great workout swimming and climbing too. Good luck.