You are powering just the booster - which is receiving/re-transmitting the signals from the sensors... According to their website -
TireMinder Signal Booster Installation Instructions -the draw at 12v is under 25ma... or .025A. A 1A fuse and capable wiring would be more than sufficient.
Quote:
Required Voltage 12v or 24v
Battery Draw - Amerage (12v) 23.7mA
Battery Draw - Amerpage (24v) 11.85mA
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The 12v trailer plug can be used for much more than just a TPMS booster - why it would have a heavier fuse.
I hooked to the existing wiring in rear to avoid that long a new run... The low draw is also why I didn't worry about it always being on (when battery is in USE). I would expect you can go lighter than 10 gauge with the much lower amperage - but certainly don't want to go too thin. I expect 14 would be more than adequate...