Hi S.,
I pumped for ten months, and I would always let my daughter finish up a refrigerated bottle later on in the day. She never had a problem. We also warmed the milk in warm tap water. Fresh (never-been-frozen) milk is a live substance filled with antibodies. There should not be any bacterial issue from rewarming.
One thing I did was to store the milk in a separate bottle and have the babysitter pour only an ounce or two at a time into her bottle. She would then pour in another ounce if the baby wanted more. If it's too easy for the baby to get a lot of milk out of the bottle at once, he might begin to prefer it to the breast. You want him to have to work at the bottle, because he has to work to get the milk out of the breast.
I hope this helps!