FYI since you mentioned the reason you want to bring beverages on board is because you don't drink caffeine. I stay away from caffeine as well.
On the Lido deck you'll find free fruit punch, lemonade, juice, milk, and water, as well as decaf coffee and hot chocolate. The tap water in your bathroom is also safe to drink (and delicious) as it is simply desalinized sea water.
Hope this helps!