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)

If you're going to be using ERG* mode primarily I suspect you could have a working solution today by staying in a single gear on the rear.

This assumes that a wider 10s chain will be ok on a 11s cassette, which IMO is likely in a single speed application. If not I'd suggest using the largest cog on the cassette to (completely?) mitigate the potential for issues. (may require using the barrell adjuster and/or lower limit screw to align it).

Next level (but still less cost/hassle) is a 11s chain, your front chainrings and rear mech should be ok with the thinner gauge. Use quick-links for the 11s chain, don't ride out of the saddle (thinner chain quicklinks warn against re-application). With a bit of faff around the FD you could bundle up the 10s chain out of the way with zip-ties (I.e clear the front chainring, but don't remove it from the rear frame triangle).

I am a Kickr user, and 8s, 10s and 11s across Campagnolo, SRAM and Shimano so I've realised compatibility issues aren't as concrete as parts of the internet suggest.

*ERG = mode where the Kickr provides virtual gearing for you such that you're outputting a constant wattage that your training regime warrants. IMO it's what really makes a smart trainer 'smart'.

You'll barely notice switching from 10 to 11 on your Trek. In general groupset upgrades aren't worth the cost and effort. The difference in shift quality between 105 and Dura Ace and 10 - 11 is barely noticeable.

I'd recommend just buying a cheap 10-gear cassette and throwing it on the wahoo - cheapest and easiest option.