The TST 507 stays awake. It takes a few minutes after turning the display on for all tires to register but they all do. Don't know what problems you had but perhaps you could elaborate in case other systems behave similarly and you would not want them.
I think the "sleep" thing is for car tires so their batteries will last many years. My Chevy sensors apparently have an internal switch so the tires need to be spinning to register.
Some RV sensors have pressure switches so they'll go to sleep when taken off the tire. TST does that as does RVi Tire Patrol but the batteries for the RVi sensors are not user replaceable.
RVi says their batteries will last two years if you take the sensors off the tires or one year if you don't. Then you get to spend $16 per sensor to replace them. But they will let you make installment payments for the new sensors...
https://rvibrake.com/products/tire-p...ry-replacement
Me, I just slip a new CR2032 in by myself every 18 months as precautionary.
Ray