I have an SQ-A, and so far I (thankfully) have no first hand experience with this problem, but it sounds like an internal electrical problem. As I understand it, the SQ-A shutter is mechanical, but electrically timed, I believe it fires at the fastest speed with no battery. The electonics catch the shutter as it opens and hold it for X amount of time. The T button causes the shutter to stay open when tripped and is released manually, so it should work.
My first SQ-A body could sit for months and the battery was still OK, it later developed some sort of internal mechanical problem. Two subsequent working bodies purchased via E-prey arrived with batteries in them, age unknown, which still worked.
I do not use any metering prisms, so the battery only is used for the shutter.
DaveT