How to know if battery in TPMS sensors has gone bad?
 

I have a 2006 Mercedes SL500. About a year ago I bought aftermarket 20" rims and tires from Discount Tires. The inner wall of the tires disintegrated several times even though I have not been driving aggressively at all. I had flats all the time (lucky Mercedes has free roadside assistance!) and finally Discount Tire admitted the tires were wrong for my car and offered to replace them. I was however given inconclusive info as to whether it was the size or the load index which had been wrong. They then gave me the car back with tires that rubbed against the suspension (amazing how they dont check that before they deliver the car). Then finally they found a third type of tires which fit. It is their cheapest tire, a whitelabel I think. However, a problem is now that the TPMS low pressure warning light is on even though the pressure is right and I have driven much more than 10 miles at more than 15 MPH. I have the version of TPMS version which gives only the warning, not the exact pressure nor for individual tires. I took it back to Discount Tire and they said the battery in my sensors must have died at the same time as they were replacing my tires, and tell me to buy new sensors at my own expense. I did see them holding an instrument next to the tire.

My question: Would bad TPMS batteries really cause the system to trigger the low pressure warnings? I would have assumed that a dead TPMS battery would instead trigger the "TPMS inoperative" message. I have gotten that message every once in a while, apparently caused by radio interference because it always happened at the same location, near an army drone testing field. But after that it has continued to work just fine.

(If someone can confirm the exact version of sensor to get, that would also be great. The car is California spec and has the panic button)

Update: One of the new tires had leaked and was empty. So I visited Discount Tire again. They said it was because the valve stem (not the valve cap) was not rotated on tight enough.

I asked them to scan the TPMS sensors. He did so and then said that all scanned good. So why then are they saying that the reason for my TPMS warning coming on, is that I need to buy new TMPS sensors from them?

Could they have installed the wrong type?