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If the sensor is broken then the dash will light up, there is nothing anybody can do about that other than getting a new TPMS sensor and paying the dealer to program it.
I hope this local shop guy takes it on the cheek and just pays for whatever lexus/toyota quotes you.
Each of the TPMS sensors has a unique ID code on them for the on-board computer to recognize. If one becomes damaged and needs to be replaced, the dealer who knows the software has to go in and delete the old sensor ID and enter the new one otherwise the car will not recognize the sensor.
If you have a new tire pressure sensor, it needs to be registered to the car's ECU. I don't know if the ID is printed outside but it can be detected by a handheld OBD and your tire shop shall have it.
My tire shop programmed 4 new tire pressure sensor IDs (Non Lexus brand) and the check system warning lights and message went away. So it is not necessary to go to the dealer. Just remember to reset/initialize the tire pressure (button located on the panel underneath the steering wheel, check your owners manual) at the end. https://www.clublexus.com/forums/5232696-post591.html