If you can find mechanical switches you like that is your best option. Lacking that, nothing stops you from ripping out the electronics and putting in the controller from a mechanical keyboard. It would be a lot of work to trace the circuit board and figure out where to jump in (and what to cut out), but any mechanical keyboard controller should work (QMK seems to be the most popular today)
You seem to not have read/understood OP's problems. He stated mechanical switches are not what he wants. Now if you can suggest switches with a membrane feel that would be helpful - I'm not aware of any but I don't know everything.
I figured that if they have the same feel as the membrane, them being mechanical is not relevant, as OP never gave a veto on mechanical options. I looked up the estimated travel and actuation pressures for those offerings in comparison to the Logitech.
AFAIK the keyboard in question only uses USB for charging. It can't send keys over the wire.
That said, before getting the MX keys I bought a Dell 7440 keyboard on ebay for $20, fastened it on top of a 6mm clear acrylic plate with M3 screws, along with a Teensy chip running kbd firmware:
If a keyboard has both wired-USB and some wireless technology, it's conceivable that physically disabling the wireless module(s) would be allowable.
I guess there are two possible hurdles: (1) getting permission, and (2) finding a keyboard in which I can disable the wireless module without causing too much havoc with the rest of the electronics and/or with the firmware.
Thanks for the tip. There might be two issues with that particular model:
(1) It's out of production AFAICT, and (2) one thing I really appreciate about the MX Keys is the indented key caps.
IIUC, if I'm willing to put together my own keyboard, I can ensure it's wired-only and I can have the key caps that I like.
The only remaining challenge at that point (I think) is finding switches with a similar feel. This is the part that's tripped me up in the past:
I've bought a few switch testers (e.g., [0]), but all of the switches I tried were way more clicky than I wanted. I really like the travel-distance, gentle landing, and relatively quiet action of the MX Keys.
I love this keyboard's feel, but I need to switch to wired-only for $reasons.