If you have not removed the transmitter ever from its original packaging, there's a magnet in there that keeps it from activating the battery. I don't have the G5, but I use the G4.
Been using my G4's until they expire for good - battery dies - but buying them every 6 months, so I have long passed the expiration dates on the ones I have. Last one lasted just over 8 months from when I started using it -- maybe it woulda lasted longer "new" but that's still a good run.
I'd say, get a new one, but give this one a try, because in all likelihood, you'll get most, if not all of its 3-month life out of it. The age does not affect its accuracy from my experience. Worst case, you'll have a new one coming to replace it.