My wife and I are planning... a trip to Bavaria. We would fly in and out of Munich and have a rental car for the duration of the trip
"our first trip to Europe"
That's obvious. It's a common misconception amongst Americans that Europe is just like the US, except smaller. Here, public transportation sucks, and we have to rent cars to get anywhere, so it must be the same in Europe, right? No, it's not the same, certainly not in Germany and Austria, where you're planning to go. Public transportation here, particularly by train, is excellent. For me, one of the great things about traveling in Germany is experiencing great public transportation, that's so much better than over here. At least for me, traveling in Germany by train is about the most relaxing way to travel.
I've spent about 170 days traveling in Europe since 2000, mostly in Germany, and never needed to rent a car. And it's not just big cities; eighty-five percent of the towns I've stayed in have less than 20,000 people, and I've never needed a rental car to get to them. You won't need a car to get to any of the places to which you are planning to go. After every recent trip, I've looked at how much I spent on rail and bus tickets and compared it to the cost of a car rental, and a rental car has always been much more expensive than the way I did it.
(I have a lot more to say, but I need some time to look up information, so I'll keep updating this post.)
Added: I think you should get a Deutschland-ticket for your travel in June. They cost 63€ ($73) for each person for a calendar month and would cover all regional trains and local trains (S- and U-Bahn, trams, and buses in Munich), including by regional train from/to Munich to/from Salzburg or Berchtesgaden. For the last days in May, you're probably best of just using local tickets in Munich. The Deutschland-Ticket should cover all your transportation, at least in Germany after May 31.
On day 1, when you arrive at MUC, you take the S-Bahn into Munich downtown for 15,10€ per person. This would be a good day to explore Marienplatz, maybe visit the Frauenkirche, maybe have dinner at the Hofbräuhaus, all within walking distance of Marienplatz. If you need public transportation to get to Marienplatz, get a Group Day Pass (Gruppe Tageskarte), for 32,60€. That's only 2, 40€ more and adds use of MVV transport for fany and all travel that day in most of the MVV area.
The next day, maybe, you go out to Dachau. Take the S-2 from anywhere on the S-Bahn trunk line down the center of town to Dachau station and take the bus, #726, from in front of the station to the Memorial. That and back is best done with a Gruppen Tageskarte (Group day ticket) for 21,20€. After that, the day ticket will cover any additional local transport the rest of the day, after you get back from Dachau.
More:
So, that's 32,60 plus 21,20, less than 54€ for the first two days. Then the D-ticket kicks in. You paid 126€ for it. That's a total of 180€ for you trip, which will cover every thing except some local trans in Salzburg and a day trip to Hallstatt, which I don't recommend (I think a day trip is far too short for Hallstatt. At least do it overnight or don't bother.)
I doubt that you can rent a car, and pay for fuel and parking for that time for less than 180€.