Is there a halachic problem with renting a hall that belongs to a church for a bar mitzvah?
There are Halachic problems with entering a church, and even with using a former church building for some other purpose. (Rabbi Frand has a tape about converting a church into a synagogue.) So if the social hall is part of the church building, that's one whole set of issues.
If it's just a free-standing social hall that happens to belong to a church, we might still be concerned about your money going to a church? While a complicated issue, the Mishna says we support non-Jewish charities too.
There could be additional problems simply with what message we're sending.
In short, it's complicated enough (many different opinions about some of the above issues; much may also depend on what type of church it is) that you really need to ask a rabbi. But I hope this helps describe some of the issues involved.
Unfortunately I do not have any specific sources I can think of (which if I can fix I will certainly try to add them) but while I'm uncomfortable with the idea of financially assisting a church even indirectly, it seems difficult to say there is a prohibition against purchasing property from a religious institution/church. It isn't obvious to me why renting would be inherently different.
Entering non-worship areas of a church is problematic as I recall but more than likely the type of hall being discussed is an independent building at a separate location. As such it would see by being willing to rent it out that they have not "sanctified" it but treat it as a secular space. As such it would seem that it might be much less problematic.
Of course there may be the issue of ma'aris ayin if it is widely know to belong to the church or is called by its name.
Of course Shalom's advice that a rabbi needs to be consulted rings particularly true with this question.