I have been approaching this argument from a slightly different direction and thought I might run it past everyone here for comment.
Essentially, the anthropic arguments are an attempt to explain constants, attributes, and qualities the universe seems to adheres to (ignoring for a moment that these mathematical representations are simple models to describe reality). Theists conjure a deity to explain these constants, attributes, and qualities, but they also grant their god with qualities and attributes which also should require explanation. The most often retort to this question I have come across is "we do not have to explain the explanation" to which my response has been "then we do not need to explain the mathematical explanations of reality. Check mate."
I am not sure that inventing an entity to explain some unknown, which itself has a laundry list of unknowns (which almost by definition can never be known) helps in solving the central issue. Once again Occam's razor slices away god.