My husband is 6'4" and similar weight. His big quarter horse handles his weight just fine. In fact, when we got him, he was the most shut down, non-forward horse you could imagine. He was owned and ridden by an anxious teen girl before we bought him. She sold him to us because she couldn't get him to lope in an arena and he was very bracey and hard to turn.
Although my husband is a beginner, he is athletic and has really worked hard to gain knowledge and skill to ride lightly and in balance. He has turned his horse from a stiff, reluctant, hard mouthed mess to a light, willing partner in 9 months. This horse has always been forward on the trail with my husband, has never indicated he wanted to quit any ride, and is now so forward in the arena that he is offering to lope every ride, when his former owner couldn't get him to lope without heavy use of spurs and crop.
It's not about the height of the horse, but more about the bone structure and muscles, so beware of only looking for tall horses. Look for a horse with a shorter back, as opposed to a draft with a long back that was designed for pulling heavy weight instead of carrying heavy weight. A big quarter horse or a quarter horse draft cross might be a good choice.
I think it will be impossible to find English lessons for him, as there is much more weight and size bias in that world. Western riders (in the Mountain West, anyway) are more about hard working horses. I know many ranchers over 250 pounds who ride quarter horses all day long. Check out the pick up riders, team ropers and steer wrestlers at any rodeo and you will see very large men on quarter horses working very hard.
I think it is very normal and prudent for dude ranches and trail rides to place a weight limit on their horses. They have no idea how fit, balanced, healthy, strong or flexible riders are going to be, so it is one way to set a safety limit for riders and horses, as well as staff who are trying to get total beginners up in the saddle. There is a big difference from a tall man who weighs 270 pounds and a 5 foot tall, elderly woman who weighs 270 pounds. They don't really have any other ways of screening people for ability to safely mount, dismount and stay on a horse.
I would look for a Western barn that actually uses their horses for work, and inquire whether they have horses that would be suitable for your husband. My husband was not keen on lessons, but watched me and my daughter in lessons for a few years, so he understood a lot of the principles of riding. My old trainer encouraged him to ride one of her horses a few times. We knew the horse we bought for him, and he has progressed slowly, which has been good for both of them.
It is possible, but it may be challenging to find the right place for him to learn.