Jay,
Based on the information you provided, weight loss will significantly improve your ability to climb harder grades. As David astutely points out, just practicing to climb will also significantly improve your climbing.
So to answer your question, doing both will likely get you there the quickest, and they are not mutually exclusive training goals.
Diet, not exercise, determines weight loss. For a variety of reasons, adding exercise to diet doesn’t necessarily lead to more or less weight loss.
I would suggest: 1. a caloric deficit diet high in protein to slowly reduce your weight while maintaining muscle, 2. no aerobic exercise, and 3. practice climbing 3-4days per week. Adding some basic finger training, like the programs found free on UA, would also be useful (due to your weight you may be at a higher risk of pulley ruptures). I would only ‘try hard’ in the gym like 25-33% of the days you climb, and just practice and have fun the other days–maybe incorporate some drills. Would shoot for a body fat ~12% as an initial goal, and not a specific weight per se. Many scales provide this metric, and it’s decently reliable, at least as a relative value over time.
Of course, if you have the time, which I think you implied that you don’t in your original post, then adding some aerobic training would also be fine, but keep in mined its to develop aerobic fitness and not for weight loss per se.
Best of luck.
— Steve