I found a crude solution for this. If you work in an office (or go to a shop lol), you can get a standard size stationary eraser and just keep it on the mat. The eraser and iPhone 13 fit together within the grooves of the mat like a glove, and you can then keep the iPhone pinned to the right.I find that if it sits on the right of the mat (UK spec) then it charges but go around a right hand corner then it moved to the left of the mat then it stops. Mine also gets really hot and super laggy - 12 Pro.
I personally find that having camera side toward the bottom gives the best results, however I still get a problem with overheating so not ideal for long journeys.
hope it helps though