If a spouse of a Medicaid recipient sells their home and buys another, do they have to pay Missouri Medicaid half?

He will have to cough up half to Missouri, and that spike in income will cause Medicaid to stop until it is spent down below $2K again.
My guess is he really does not want to move anyway.

If its a lateral asset exchange, it may not matter as the base amount for assets remain the same. Like farm with 259k tax assessor value then house purchased must be 259k assessor value. If its a home for lesser value, then the difference will affect his medicaid eligibility. I would get an elder law attorney on this, as there likely going to be issues at a secondary review level of Medicaid on this, and you want this to go as smooth as possible. Also there could be issues with how value is asssessed.

What also could be good is for her doc to do an "advisability" letter, like " as the MD of Jane Jones I have advised her to be within 15 minutes of an ER due to her brittle diabetes....."

New home will still be subject to MERP but if she survives him, then she will need to file the MERP exemption as surviving spouse. If she predeceases him, then house subject to MERP. Speak with attorney if any options at all feasible to keep MERP reduced ( she wills her half to someone other than hubs)

Just a tidbit on the " lateral exchange", insurance companies pay close attention to this on policy pay outs. We had friends who had homes destroyed by hurricane & had windstorm insurance. 750K policy pay out, this was a pretty expensive specialty premium policy too. They got 250 max flood also. They bought an interim home for 250 further down on the Gulf and planned on holding the rest to use to rebuild back on old house site. Within the year, they got a demand letter from insurer for the 750..... They did not "replace" but downsized & not allowed under their policy as it must be for replacement. They ended up buying 1M home so to keep the funds. We only got our full windstorm (& 2 years later too) only because we rebuilt and provided insurer with building permit and builder contract that was OVER the replacement value (insurer did a 25% depreciation). I'm digressing, but my point is that you have to pay attention to what the powers that be use to set value. They could look at the value of the ranch as a separate value from the value of the home on the ranch. Ranch isn't being replaced only the house is, that could be an issue. All this totally do-able but probably with a flurry of paperwork. Get good legal. Good luck.