Why do you need a whole single thing to do it?
The current SUV can be programmed to know 2 sets of five (for summer + winter sets), and you change between them via a switch under the steering wheel. It's common for new owners to get TPMS system errors because the switch got pushed by accident.
Because auto-learning ones don't encourage a trip to the dealer, where the customer will go for new tires as well etc.
Over fifty degrees (F) you'll see 5psi change for a passenger car tire (30-40psi.) A 10 psi difference requires a 90 degree differential.
That's a change of between 12.5% and 33% depending on tire.
Additional load doesn't change the total volume much - so it's not directly increasing pressure - but it can increase deformation (as you pointed out - contact patch size changes) creating more warming, which will increase pressure.
As far as diagnosing and programming at home, I have a handheld OBD-II reader ($35) that can report TPMS codes like this from the car (without a laptop), and also an OBD-II TPM QuickSet ($120) from ATEQ (a more basic model than the shop had). The QuickSet tool is configured over USB, then attaches to ODB-II to write new TPMS codes into the car. It's useful for swapping between summer and winter wheelsets without a shop.