Wahoo Kickr cassette question
You're right - something has to change.  11 will not work properly with 10, and vise versa.
Ideally, you're going to have to standarise on 11 speed on both bikes.  That means new shifter, chain, cassette, and maybe rear derailleur.  This is probably the most expensive solution, though you can sell off the 10 speed parts to recoup some funds.
Your other option is to move your 10 speed cassette to the trainer for your bike, (or have a second 10 speed cassette spare)  and swap an 11 speed cassette onto trainer for the other bike.  THis is faddly and will add wear and inconvenience.
Your third option is to find a third bike and leave it on the trainer permanently to share, though that bike is unlikely to fit both of you.   Or buy a second trainer and configure for 10 speed (though this is a ludicrously expensive solution)