How do you break it to Mom she has to go to ALF?

In this case, you do it matter-of-factly. Because it is a matter of fact, with dates and logistics attached to it.

When is your surgery scheduled?

You sit down with her and the family diary, and you point to the dates and say "this is when I'm going to St Saviours, this is when I go to rehab, this is when I should be recovered. Meanwhile, this is when you are going to stay at Sunshine Horizons and you are booked in for x weeks initially*."

That's plenty of information for her to absorb in one go. She will have questions and arguments: you answer those which have a sensible answer, and pat her hand reassuringly in reply to the others. Do you know which ALF?

* this is not a lie. It's the truth, but there can't be a whole truth because nobody can foresee the future.

Honestly? Since she has already burned through the goodwill of your siblings and you admit she is a narcissist the chances of her accepting this in a reasonable way seem pretty slim, I think I would present it as a fait accompli - as lovingly as possible of course - and stand back for the fireworks.

Just like you told us. You are going to be incapacitated for several months or longer. You need to get her situated now before the surgery and try to get things set up for her benefit as well as your husband’s. If there is anyone else who can be her advocate, now is the time to get them involved.
Don’t neglect to get your own paperwork in order.
Good luck with Medicaid and with your surgery.

Kindly but firmly tell her what's going to happen. She may not like it, but that's too bad. You have to do what you have to do and take care of yourself first.

Just tell her - straight out - you are going to have surgery and will not be able to take care of her during that time. That she will be going to a "senior apartment" "temporarily". Reassure her you plan to visit as much as possible as well as notify her friends, church, etc. of where she is and the facility will make sure she gets to her doctor appts, etc. I would even move her in ahead of your surgery so you can help her get settled.

Trying to explain reality to someone with dementia is a daunting task, because they have lost touch with it. This is where the “therapeutic fib” comes in. Mom will not understand that you need surgery and your husband cannot take care of you, her, the dog and the house. You can tell her things like men are coming in to “work on” the house and she needs to go someplace for “a while” while the work is being done. You know her the best, and you know what to tell her. Be kind but firm that this is the way things need to be. Do it ASAP so she has some time to adjust and you won’t have to deal with this while you are recuperating from your surgery. Good luck. Sending best wishes for your surgery. Come back and let us know how you’re doing.