It is possible to be infected with false Paleblood before becoming a Hunter, thus effectively making you a false Paleblood Hunter, yes... though it will probably be a lot more challenging. The batch of false Paleblood Hunters currently under metamorphosis or in the currently active scene were created en masse as a test of the treatment, to see how well and reliably it worked across a broad range of subjects. Anyone receiving the treatment to become a false Paleblood Hunter henceforth would do so [i]after[/I] the procedure had been tested, meaning that the Healing Church would already know that it has a high probability of working, which in turn will mean that the church will be a [i]lot[/I] more selective about who they decide to effectively turn into immortal soldiers.

A character would need to either be very highly favored and deeply trusted by the Healing Church to receive such an offer (as opposed to just being made a Hunter, which they would be much more eager to accept), or they would need to steal the means to infect themselves with false Paleblood and then become a Hunter.