The Zwift cog could certainly be a solution, as long as you have workarounds to shifting gears on the cassette. I have the cog and adjust the trainer resistance on my garmin head unit. Others may use ERG mode, or on Zwift, the electronic shifting.
I expect the 11sp rim brake bike has QR type wheel and dropouts, and the 12sp disc brake bike has a 12x142mm thru-axle. To switch between those two, the trainer needs to have the hub caps switched. The QR caps have 9mm dia protrusions that the frame’s dropouts sit on and uses a QR skewer to clamp the bike to the trainer. The thru-axle caps have a 12mm hole through them that accepts the bike’s thru-axle which passes through the left dropout, the hub, and screws into the right side dropout. Then there’s the issue of the cassette. If not using the Zwift Cog, the trainer will need a 12sp T-type compatible cassette. This cassette will work with the 11 speed bike as long as you leave it in one central cog and don’t shift the RD. This works OK for ERG mode, but not sim mode (like Zwift, Rouvy, etc.). Usually it takes a little adjustment (cable tension or electronic micro-adjust) to get the chain to line up perfectly on that cog.
OP didn’t mention having with Transmission, so may not need a T-type cassette.
I’d go with a Zwift Cog. Anything else would not meet the desire of “minimal” change. If your 11 speed is a mechanical drivetrain, you could index the shifters so it was smooth on a 12sp cassette and set the limit at the 2nd lowest or 2nd highest cog.
I just ordered a Zwift cog today at a black friday discount. My idea was to achieve just what you are aiming at. Currently I have an 11s rim brake bike setup on my wahoo Kickr. My 12s disc brake bike is currently in stand by if weather here in Sweden suddenly improve so I can ride outside. I’m not very optimistic about the latter, so why not give it a go. I can give you guys an update on the project some day next week if you like.
I have used a 11spd shimano casette on my trainer with a 12spd Sram Flat Top Chain for the past 3 years - I just did‘t shift and used ERG. Fourth or fifth cog worked best. You still need to change the endcaps but this would be the most cost effective solution since you already have the parts I assume?
I guess you have got plenty of answer at this point so I don´t think I can add much to the discussion. Anyway, in my experience a flattop SRAM 12s chain works reasonably OK on a 11s Shimano cassette if you keep it in a single gear somewhere in the middle of the cassette and stay in ERG-mode. I didn´t experience any significant difference in noise level between the two but the cog offers fine tuning of the position of the cog in relation to the derailleur, which might be an advantage depending on your setup.
Virtual shifting could be a benefit if you are running Zwift. Allegedly Rouvy supports virtual shifting, but it seems like Zwift is actively working against compatibility with other platforms.
Bottomline: Unless you are on Zwift i doubt there are any major advantages to to use Zwift cog over a standard 11s cassette, at least for now. Unfortunately that limits you to ERG-mode.