Can a father use maaser money to pay a mohel to give his son a bris milah?
The question is strenghtened since a person is not allowed to charge for performing a mitzvah.
One can't use maaser money for mitzvos that one is obligated to do, including circumcising your son.
There is a Machlokes as to what one may use his maaser money for. The Rema (YD 249:1) writes that it may only be used to support the poor and not for any other mitzva, while the Shach (249:1) quotes the Maharam who allows one to use one’s maaser for a mitzva that one would not have otherwise paid for.
The Chasam Sofer (Shut Chasam Sofer YD 231) explains that if one initially intends to use one’s maaser for other charitable causes or mitzva expenditures, then one may do so. This does not apply to Mitzvos that one is obligated to do, such as buying a Lulav and Esrog. Thus, one may not use one’s maaser for Machatzis Hashekel or Matanos Laevyonim. The Magen Avraham (OC 694:1) and Mishna Berura (694:3) write that once one has fulfilled the basic mitzva (by giving 2 coins to 2 people, Mishna Berura 694:2) one may continue distributing money from one’s maaser.