Well, at my parish church we also use a Shure radio mic, without problems unless the battery is allowed to run low. It has a Lemco connector, a make with a good industrial reputation, which latches in place so mis-seating shouldn't be a problem.

Does the receiver indicate the received signal strength? Some do (e.g. on a row of LEDs), some don't. If possible, when the problem occurs, check whether the signal is low. I assume you have a reasonably direct line of sight from transmitter to receiver, so that the signal doesn't (for example) have to go through several feet of stone wall.

Another possibility might be static, in the more literal sense of static electricity rather than atmospheric EMI. Are the ministerial vestments made from artificial fibres by any chance?