A6 tires- 1 blown

Last night I hit a pothole and blew a tire on my 03 A6 2.7T. The set of tires is relatively new but the tire store would not put one tire on. They said I had to replace all 4 tires or else it would screw up the all wheel drive, which will lead to a much more costly repair. The difference between a new tire and the existing 3 are 4/32's. I live in an area where we do have snow and rain regularly. Is this true? If so are there any good tires that don't cost $800? Thanks, Ken