Welcome to the RTA Forum!

Your plan is not possible, because CA-120 through Yosemite (Tioga Pass) is closed all winter long.

If your only reason for going to Mammoth is for winter fun, I would skip that location and look to do your skiing/snowboarding elsewhere. The Lake Tahoe area would be one very obvious location - you can go north from Yosemite, and the cross the Sierras on one of the year round roads like US-50. There are also several ski areas on the Western slope of the Sierras, including Badger Pass located within Yosemite NP.

I would not recommend a Mustang for this trip, as you do not want a rear wheel drive sports car in the snow/mountains. You are also correct that switching cars is likely going to be much more expensive. The best value is always going to be a standard sedan, although perhaps you'd want to consider an SUV for this trip considering all the mountain driving you'll be looking at. Rental cars will have all-season tires on them, not snow tires.