Don’t worry- my barn doesn’t have a huge wallet either:0
have you tried a grazing muzzle? They cut down on grass intake, but do not restrict water intake. Your local Tractor Supply or something like that would have one:)
Also, check your hays’ sugar content. Your hay may have excess sugar which is helping your horse gain weight. Check what kind of hay it is too. Alfalfa has excess fat than timothy or orchard grass.
Maybe in addition to dressage you could occasional jump your horse, if that is a option. Also try hillwork, trotting up hills, walking down hills, speed hills. They should build muscle, and eliminate fat.