If it were me, I would give it a try. Maybe set up a time limit with your cousin- say "we will do this for xx months" and reasess the situation at that time. If it is working for the 2 of you, continue. If not.. it is not like you dont have the option of bringing your horse home. It is nice you have that back up.
That said.. are you comfortable with your cousin riding your horse? Do you agree on all the details like how you expect your tack to be taken care of, will she make sure to pick his feet etc? What if she gets bored or tired of it? (she is hitting the age when boys and school activities can take over..) What if ONE of you likes it, and the other does not? Can you afford to keep the horse there if she backs out?
I have always boarded my horses, and no, they dont get sick. Never have. But the BO may not always do things the way YOU would. Your horse will be with others, and of course they need to establish a pecking order. You will want to be prepared for that. Expect some nips and kicks if they are turned out with others. If you arent used to that it can be a bit alarming. Barn drama exists.. but not everywhere. If it is a problem, I would just try to keep my head down and enjoy your horse, and not get sucked in.
WHy not give it a chance for the summer, when your cousin is able to ride and be with the horse a lot, and see how it works out. If it doesnt, bring him home. Not like you need to make a 'forever' decision...