Look up train connections on http://www.bahn.de. Switch to English. Tick "only local transport" to fiund those connections the Bayern-Ticket is valid for. The direct RE trains take two hours.

Train travel requires a little planning, which is something car freaks are not used to. On the other hand, I don't think you'll be much faster by car. Besides, there is the environmental aspect. Then the train allows both of you to relax - by car, the driver has the stress on crowded roads. And renting a car plus petrol, parking fees and everything will cost maybe 3 times as much as the train ticket, maybe more - treat your wife to a nice lunch in Salzburg, and later coffee and cake in a fancy café, for that money...