Few points:
1) Weight training vs. cardio: Neither of them make much of a difference in caloric burn, unless you are at a fitness level where it probably doesn't really matter (e.g.,. if you are hitting the climbing gym 5 days a week, 3 hours per session, plus outdoor climbing, caloric burn is gonna go up).
2) Where weight training helps is in making sure the weight lost is actually fat, not lean tissue. So sure, add some SE to your plan.
3) The sad fact of weight loss is that burning calories makes you hungrier (for most people: certain rare freaks have an appetite loss). So weight loss ultimately comes down to the intake side of the equation.
4) Protein is king.
As
jessebruni wrote, the best diet is the one you stick with, and weekly caloric deficit is key. As
jonfrisby wrote, you can control consumption/intake - weight loss is an output. So therefore, give yourself a plan that you can stick with, that creates a weekly caloric deficit.
Here's a framework: say you have a long term goal of being 175# at 6". Break that down: get to 197# first. That means you have 20# to lose. Give yourself 1"/week -- means you need a weekly caloric deficit of ~3500 calories. So you get to eat an average of 2200 calories a day* or 15,400/week based on your intermediate target weight.
One way to get there is the bodybuilder approach: plan all your meals so that you eat more or less the same thing (chicken breast, brown rice and steamed broccoli for the bodybuilder classic meal). For some people, that's the easiest thing to stick to.
Another way way to get there is the 5/2 intermittent fasting plan. Take two days a week (rest days, please) where you eat ~500 calories (all protein and salad), and eat whatever you want the rest of the days.
Or maybe you always blow your diet on the weekends? So get super strict Monday-Friday and create a larger deficit to create a buffer. Ideally, do your hardest climbing and weight training on the weekend days where you have a larger caloric intake.
If you take that approach to the extreme, you get a cyclic dieting approach, where you eat like a mouse (well, a mostly protein eating mouse) on rest days, and eat normally on training days.
But it all comes back to:
1) Control intake
2) Create a weekly caloric deficit
3) Use a plan that you can live with.
* If you look at an online calorie counter, you'll see that I'm rounding down. This is because the human homeostatic system basically sucks for weight loss, and because when you use calorie counters, you will lie. Worst that will happen is that you'll get to your target faster.