Howdy and welcome to the forum
I'm unsure how the whole 'weight limit at lesson barns' thing works, though I'm sure you can find a place without too much difficulty. I've heard of some barns putting a 250# weight limit, but you would probably be able to get around that by maybe taking private lessons on, perhaps, a lease horse or borrowed horse. The only way to know for certain would be to call around to barns and find out if they have a weight limit and, if they do, what you can do to find a way to learn anyway.
Now, as for finding a horse for you, you don't really need to look for drafts or draft crosses unless you just want to. The conformation and proportions of the horse matter a lot more than the actual size of them. The type of horse you would probably want to look for would be stocky, have moderate to heavy bones in their legs, have large feet, a short back, and a big barrel. Horses like that can carry much more weight for a longer time than a big old gangly thoroughbred can. I would have zero qualms about putting you on my mustang, who is rather short at only 14.3, and letting you ride him just about as long as you wanted to.
Truthfully, if you find a horse that is built to carry the weight, you really shouldn't need to worry about limiting your time on them unless the horse is severely out of shape...like if they'd been turned out to pasture for years with no riding at all. A properly conformed horse that is moderately fit should have no problem carrying you all day...though it will take a fair amount of time before you're
able to ride that long
Something to consider as well that might help you to work off a bit of weight and maybe help to pay for your lessons too, is to look for a place where they might be willing to trade lessons for manual labor. Mucking stalls, scrubbing buckets, and tossing hay for a couple hours a day can melt the weight off a person in a hurry. Heck, when I moved back home and started working at the barn all the time instead of spending my time in an office like I did at my old job, I dropped about 20 pounds in a relatively short amount of time.