A few things to try.
1. In the tire menu, change the tire size from 18" to 19" rims. Drive a bit, then change it back to 18". This will serve as a "reset". From what I recall, it normally takes 5 to 10 miles of driving for the system to relearn everything, assuming the system is OK.
2. Bring it into the dealer since it's still under warranty. The PWIS computer can run a diagnostic on the receiver unit and can tell if there is a problem with the unit or the senders/monitors in each tire.
The TPMS batteries can last from 5-10 years and yes, when they go you have to replace the whole unit, not just the battery. However, with all 4 tires indicating a problem, I don't think it'll be related to the battery.
I don't know if there is a specific fuse to the TPMS system, but if there is, pulling that can also server as a hard reset of the system.
I would take it into the dealer first. The service manager is giving you the worse case scenaro over the phone to cover all their bases. Hopefully it'll be a simple fix and no dismounting of tires will be needed!