I always leave it on. I maintain my gear and if I'm somewhere that I had to use a rental tank I pretty much always replace the o-ring on the tank first. Around 200 dives now and I've had zero problems doing it this way. I think problems come in if you've got poorly maintained gear that leaks air.
Are you sure your tx stays active when pressurized but the pressure isn't changing? My scubapro tx definitely stops transmitting after about a minute of no pressure change.
From the Shearwater Perdix AI manual.
If there is pressure it is transmitting. The only way to turn them off is to close the valve and purge.