Hiya, my DS is the same age and still having his 3 cups of milk (sometimes 4 but he's dairy free which is the only reason) he does however drink his water.
G has water on offer at all times and he will get it himself throughout the day. he has it before breakfast and also with his meals. Does yours have it during meals?
You don't want to be giving anything other than milk or water from a bottle as its bad for teeth, however I can see why your doing it if your trying to introduce something new, you just don't want it to turn into a habit cos its bot good long term
Does he get his cheese and yogurt etc making up his dairy daily allowance? If so I would stop his mid day milk. If not encourage more dairy products and stop the say time milk.
DS has breakfast then milk (when having milk first he refused breakfast)
If he gets up and has breakfast before 8 he will have a "snack" of milk at 10
He'll then have a snack of biscuit and fruit with milk at 2
Milk at bed.
So for snacks, he us yet to have food in the morning as milk is enough to last him until lunch, whereas in the afternoon he can't last until tea so he has food too.
If DS was allowed dairy, I would still offer milk at these points but less, and add food in there to bulk up the am snack.
Sorry if i'm rambling LOL
So in all....
If I were you I would cut out day time milk to encourage water or even squash (then gradually weakening it if you don't want him to have squash, but it is enticing him in), then once he is drinking fluids in the day, I would personally add a little milk for snacks BUT you don't have to, you can just keep to water or squash.
I hope that helps a little