I've been thinking about this, quiddity. It puts me in mind of my many stand-offs with my dad, who was as mad as any Genaral/President of any tropical country you care to name! (Probably would have been one, if he'd been born there instead of in Kent ...) Anyway, by his last years, we'd arrived at a state of mutual respect & understanding - no liking iyswim, but he trusted me because I'd proved I didn't trust him

After the age of 25, I never stayed at my pernts' house when visiting (I didn't visit overnight from 18-23, but that's another story). I paid for my own hotel and, as the years went by, my P&M split my room cost with me. For his last couple of years, he actually offered to pay my full cost - incredible! I've been thinking about how this happened. I reckon it's because I became intransigent ... like him. With hindsight, I could have arrived at a workable conclusion a lot faster than I did - live and learn, hey?
Here's my suggestion.
Tell her she was not invited and you have other plans.
Tell her she bought her ticket without an invitation.
Tell her you know she's your mother, etc, but there are a hundred reasons why she shouldn't travel.
Tell her you don't care if she has a heart attack on the plane, it's irresponsible to fly in her condition (even if you feel like you're lying - it's the truth, so say it).
Tell her you accept no responsibility for the outcome of her choice to fly, uninsured and therefore illegally, without an invitation.
Tell her your Xmas is so busy, you can't possibly have her to visit.
Offer a pathetic sop, maybe lunch together on the 23rd, if she's well enough. Say that's all you can manage.
SHOUT if necessary! That's what made the diff with my dad - I set really tight boundaries, and shouted as hard as he did.
Rope in all & any friends to boost your confidence.
Email bro at the same time as you speak to M (so it's in writing). Bugger his feelings, he's relying on you to take up the miles of slack.
How do you feel about this?