I asked because I know of no technical reason why a wireless system should not work. This is a chance for me to learn.
The general question "do wireless systems work with active pickups?" seems to have Yes as an answer which is not your impression.
Some active systems are "too hot' for the wireless and so the active system needs to be attenuated in some way. That is usually adjusting lowering the volume or sending the output through another preamp.
Some active systems interfere with the wireless and the solution seems to be better shielding.
Specifically on your G10 is seems as if there is sometimes a polarity issue that can be fixed by swapping wires on the guitar's output jack.
https://line6.com/support/topic/20888-attention-g10-owners/ and read until you see
I'd like to think you could make the G10 work for you although it might not be a "plug and play" solution.