Can skilled nursing facilities refuse to take a patient?

I suggest you see a lawyer concerning your combined assets. You need to have assets split. His split will go to his care and when its almost spent down, you can apply to Medicaid. At that time you will become a Community spouse. You will get partial or all of your monthly income (SS and pension) if its needed for you to live. Just giving the basics here. You need to see an Elder lawyer versed in Medicaid to set this up.

I think you mean Memory Care facility not Skilled Nursing.
Memory Care can refuse if they do not think they can handle a resident. This often happens if someone is violent or has been in the past. They typically want 90 days with no incidents.
As long as you have the legal authority to place him in Memory Care you can.
Discuss with his doctors the delusion, any anxiety that he has and there are medications that can help.
Tell him that his "wife" will meet him in his new home.

I would follow the advise of the neurologist and psychologist. He will not be an easy placement at best. SNF are usually only covered when skilled nursing care is needed for a medical issue. I don't think he would be admitted. At best he will not be an easy placement, so I would start looking at once.
As you are married you will need to check on division of assets so that all your funds aren't eaten away in your husband's care. MC is exceptionally expensive. I would see an Elder Law Attorney for your options at once.
I wish you the very best of luck and am so sorry.