The base of a Bloody Mary is tomato juice, and folks can tomato juice all the time in hot water bath so it should work. I can't help on the flavor, I don't like Bloody Mary's. It would probably be a good idea to add some ascorbic acid or lemon juice to get the Ph down.

I would bet that a pressure canner would work for hot water bath canning if you used a different loose fitting lid-or leave the lid off. Pressure canning would probably be faster and safer(bacteria wise).

This recipe uses salt and makes no mention of adding acid....Bloody's have salt in anyway?

http://www.food.com/recipe/tomato-juice-canning-188981