Dr, "pool" and "budget friendly" don't go well together if your budget is under $100/night. Over that ($100-200), the ones recommended are excellent plus several more. Under that, Ruby's Hotel in town, on the beach, but no pool. They have 4 rooms with AC (very basic rooms) @ $65 US/night. I've stayed there 3 times and am going back.
Now with a pool you'll have to go a couple of blocks off the beach @ $65-75US: google: Cheap Hotel San Pedro Belize. I have not stayed there because I must have beach and water at my front door!!