He can file for custody of his daughter. The court will look at who will provide the better environment for the child. The father's income being higher than the mother's is not likely to be an issue (it will be an issue when she files for child support, which she may get even if he gets custody.) The father does not have to prove the mother unfit, he has to prove that he can provide a better environment for the child, and that he is willing to work with the mother to allow them to develope a strong relationship, as well.
As for your other posted question on here, the father can petition the court (and should do so immediately) to order the mother NOT remove the child from the state. However, the situation you are in, with him going out of state for 3 months, is going to put you in an awkward situation. The court is not likely to give him custody of the child knowing that he is going to leave the child with someone else (you), and they will not allow the father to remove the child from the state any more than they would allow the mother to remove the child from the state (she can petition the court to halt the child's removal, just as your boyfriend would do.)
It might be in his best interest if he hopes to get custody of the child to see if he can "get out of" leaving the state for 3 months while there is a custody hearing about to take place. If that fails, perhaps he can have the custody hearing "stalled" until he comes back. (Perhaps he needs an attorney?) If he should opt for that, he should do everything he can to keep up visitation and contact with the child through the period he is gone. He should also keep good records as to his contacts with the child to SHOW that he has kept up visitation and has a strong relationship with the child.