Eric,
I think that one of the problems here is that you are using the wrong terminology in referring to the suspected defective part as a transfer switch. The part that you are referring to is called the pass through relay, and it is a replaceable component in your inverter, but it comes as part of a circuit board. I am not sure if the battery charging circuit is on the same board or if it will take two boards to repair your inverter.
A point to consider is that Heart inverters (now Xantrex) are not the most trouble free inverters, and with the age of yours, it might me a better plan to think about just replacing it with a more dependable unit like a Magnum inverter if the repairs get anywhere near the price of a new inverter.
Gerald