So I'm 6'3" and have ranged from 220-250 over the last 19 years in this business. Right now I'm about 230 which, coincidentally, is what i weighed when i graduated from high school where i was an athlete. Granted, I was an offensive lineman and the weight was an attribute. I've been lighter and heavier over the years, usually in direct proportion to how much i was working and how much beer i was drinking. I have always been able to keep up with most everyone I've worked with, and I've worked with some no-bullshit, bad-ass climbers. While I'd like to say that my weight has never held me back, it's hard to say because I've never not carried the weight. My advantage was being smarter than the average bear and creative and resourceful in how i approached my work. Over the years, I've had my fair share of injuries. Mostly knee problems and a nice new case of pretty wicked arthritis in my hands. What I've come to understand is that more than my weight, it has been my age and conditioning that have been the greatest limitations. As i have aged and my physical conditioning has gone back and forth, I can still do all the things i used to do, just not for as long and the recovery time from an injury or a hard day is much longer.
So, to the OP. Could you be an occasional climber performing simple pruning and maybe some straightforward removals? I certainly don't think that your weight alone should preclude you from it.
Should you lose weight? Of course. if not for the climbing then for your heart. And climbing might be a good way to help you along the road to better conditioning. The more i climb, the better i feel both physically and mentally. My wife always knows when I've been out climbing a big nasty tree all day because I come home all pumped up and bouncing off the walls. Conversely, when i get stuck in the office for a day she knows as well--I'm all tired and mopey.
Will you ever be a "climber," in the vein of a full time production climber? Probably not. But that doesn't seem to be your goal.
I think it would be presumptuous of any of us to give a definitive "NO," on the question and, frankly, a bit arrogant and elitist as well.