Do you have the EVIC or is it the simple tire outline on the dash? Sounds you like you have the simpler design and not the EVIC. Does one tire appear to be lower in pressure? Did you check them with a tire gauge? The TPMS (as far as I know) does not use the ABS sensor. It's simply monitors the air pressure (to my knowledge). The light and chime will sound if one or more tires falls below the 25% threshold or there is an issue with the TPMS system.
I had the stem of a TPMS disintegrate on my 2010 Journey. Temporarily I had a regular valve installed. Yes, this resulted in the TPMS light on all the time, but we lived with until I could get it replaced. Mine only dinged at start up. We were leaving on a trip and didn't have time to have a new TPMS installed. I did eventually replace it when I purchased new tires. The Firestone shop only charged me for the new TPMS ($80) since I was already paying for the labor to mount and balance the new tires. No charge for any programming.
Have you had the TPMS system checked for any fault codes? You'll probably be charged a diagnostic fee (1 hour labor).
Here's what I found in my Journey manual:
Tire Pressure Monitoring Low Pressure Warnings
The Tire Pressure Monitoring Telltale light will
illuminate in the instrument cluster and a chime
will sound when tire pressure is low in one or
more of the four active road tires. Should this occur, you
should stop as soon as possible, check the inflation
pressure of each tire on your vehicle, and inflate each tire
to the vehicle’s recommended cold placard pressure
value. Once the system receives the updated tire pressures,
the system will automatically update and the Tire
Pressure Monitoring Telltale light will turn off. The
vehicle may need to be driven for up to 20 minutes above
15 mph (25 km/h) in order for the TPMS to receive this
information.
Check TPMS Warning
When a system fault is detected, the Tire Pressure Monitoring
Telltale light will flash on and off for 75 seconds
and then remain on solid. The system fault will also
sound a chime. If the ignition switch is cycled, this
sequence will repeat, providing the system fault still
exists. The Tire Pressure Monitoring Telltale light will
turn off when the fault condition no longer exists. A
system fault can occur due to any of the following:
1. Jamming due to electronic devices or driving next to
facilities emitting the same Radio Frequencies as the
TPMS sensors.
2. Installing some form of aftermarket window tinting
that affects radio wave signals.
3. Lots of snow or ice around the wheels or wheel
housings.
4. Using tire chains on the vehicle.
5. Using wheels/tires not equipped with TPMS sensors.