Saef, nice work on your deadlifts.
I was thinking, if your gym has kettlebells, you might try the lunges with those. They're less of a pain to try and grip and would probably let you up the weight on your lunges without grip being such a limiting factor. A little chalk would help, too, if you are allowed to use that in the gym. Some gyms will allow a chalk bag if you don't make a mess with it, even if they don't allow loose chalk. You don't need much, just a thin layer does the trick.
For the wrist pain, you might try wrist wraps or a couple passes with athletic tape. I wouldn't use them for everything, but if it comes down to being able to do the work you need to do, or not, then it's like a weight belt...put it on so you can get the work done.
Grip is slow to improve, but it will catch up if you just keep working. Maybe when you're done testing out this program, you can throw in the grip work after your regular lifting session a couple times a week, and help things along.
You don't say how many reps or sets you're doing with the lat pulldown. If you can't figure how to bridge the gap with loose plates, another approach would be to add an extra set or two with the 80 lbs, or jump up to the 87.5 and do fewer reps per set (then build up the reps from there), do a set or two to failure with the 80 lbs after your regular sets...any of those would get you stronger and pretty soon you'll be able to do the next setting on the machine in the rep range that the program calls for.