I admire your resolve too, but I think this would be one mountain too far, for all the reasons mentioned - we have hired many, many cars over the years in various countries, and we‘re fine with driving on the left and right, but I wouldn’t do this.
For me it would be that most Alpine countries insist not just on snow tyres for compacted snow, but also winter or universal tyres for all winter conditions. Presumably your own car already has these installed. It’s incredibly difficult finding car hire in the UK that offers this (I tried some years back).
Car hire companies have got increasingly scammy over the years, meaning that if you don’t take out full collision damage waiver they’ll try to find some sort of scratch, somewhere, to hold onto your deposit. I always do full CDW as a precaution and just treat it as a necessary cost of the hire. And I take a zillion photos and videos regardless - easy to delete them afterwards. When checking a new hire car, if you find even the tiniest scratch that’s not listed on the rental agreement you must get it added.
Even if you find a reputable company (Sixt, Europcar etc) that will let you take a hire car that far post-Brexit, you‘ll be charged through the nose for the privilege, plus excess mileage. It would honestly be cheaper to buy a used car, which you can either carry on using or sell afterwards. If you do buy, make sure you have breakdown insurance that includes a replacement hire if it, ahem, breaks down.
Or just bite the bullet and fly/train there plus local hire or shuttle to the resort. Although driving a left-hand drive for the first time ever, without another adult in the car, and in snowy conditions, is also far from ideal.
Best of luck with it, whatever you do.