It sounds as if the cart is bothering him, probably unbalanced.
Are you using a two wheeled cart?
What might be happening is when the bride gets into the cart it causes pain to the horse. A lot of horses cannot stand when the shafts go up when someone enters from the back of the cart, causing pressure upwards on the horses' girth. Or the bride is entering from the front and causing a lot of weight on the horse's back. Neither of these occurrences should happen. I am sure that the bride is not quick getting into the cart also with the dress, veil, heels, bouquet, trying to stay clean and everything else. Prolonged pressure can be painful.
Is the cart unbalanced the other way, to heavy on the horses' back?
The best way to see if a cart is balanced is measure from ground to shafts when the cart is in the shaft tugs when hitched to the horse.
Then when the cart is not hitched lift the shafts to that level and have one person get in the cart, and then have the second person get in the cart.
Take note how the shafts move while you are holding them, how much weight is on the shafts, that is how much weight is on the horses' back.
There should be no more than 5 - 10 pounds on the horse.
To balance the cart better move the shaft tugs up or down that should make a difference, also use a mounting block, it will help the bride not to have to haul herself up on to the cart from the ground, she would be able to step on to the cart more towards the middle, (towards the axel, center of gravity).
Practice with the horse not during a wedding to see exactly what the problem is. A well behaved driving horse that acts this way is showing pain, and you are smart to realize this needs to be addressed before a good horse is ruined.