There is so much I want to say about your itinerary.
First, I have spent months, total, in this area without ever needing a car. Public trainsportation (Bus & Bahn) in southern Bavaria is excellent. I really think you can do this trip without a car, and for a lot less money. There might be time or two when you can save a few minutes with a car, but so many of your days a car will go unused or under used that it will not pay for itself. For most trips on you itinerary, a 31€, 2 person Bayern-Ticket will sufice. Your children will ride free on full fare and regional tickets and regional passes.
Mittenwald: I've been two times to Mittenwald, once overnight, once for an afternoon. It's a charming town, but I don't think it is worth 3 nights unless it is used as a base for other places. It's only 25 minutes by train to Garmisch-Partenkirchen, a little over an hour to Oberammergau by train and bus.
Oberammergau: Same as Mittenwald. I've spent 3 nights in Mittenwald on two trips. Again, it would be good as a base, but not as a venue by itself. It's 40 minutes to Garmisch-Partenkirchen by bus.
Garmisch-Partenkirchen: Much as I dislike GaPa for it's size and touristism, it's at the base of the Zugspitze, and you want to go up the mountain early to avoid the clouds that often shroud the summit later in the day.
Of the three, I might pick Oberammergau. It's close (1½ hr) to Füssen. After visiting the castle in the morning, you could catch the 4:10 PM bus from Füssen and be in Oberammergau just after 5:30 PM. Use an RVO Familien-Tagesticket for 24€ for the 4 of you. The next morning you could catch an early bus to GaPa, go up the Zugspitze, then go on to Mittenwald in the afternoon. The folowing day you could take in Linderhof (about 35 min by bus from O'gau). I'd skip Ettal; the main attraction is just another over-the-top Rococo church of which there are so many in Bavaria. For each of the two days in the Oberammergau/Garmisch-Partenkirchen/Mittenwald area, use a Regio-Ticket Werdenfels for 22€ for the four of you.
For Oberammergau to Berchtesgaden, a car could be two hours faster than the train, but you have time. With a train, you wouldn't have to worry about traffic jams on the Autobahn. With a Bayern-Ticket (again, 31€ for all of you), you can use any regional train. You could stop for an hour at the Munich Hbf and have lunch at the food court there.
Once you get to Berchtesgaden, your accommodations should give you a guest card good for any bus transportation in the Berchtesgaden area. You can use it to go out to Köigssee or up to the place where the buses leave to the Eagles Nest. Regardless, if you go to the Eagles Nest, you will have to pay extra for the bus up the private road. The Dokumentation Center is right there at the startup to the Eagles Nest.
If you plan to spend a day in Salzburg, you will need another RVO Familien-Tagesticket for 24€. I would definitely spend three nights in Berchtesgaden. One for the Salzburg trip, one for the Eagles Nest/Dok Center, and one for Königssee.
For the trip back to Munich, use another Bayern-Ticket. It covers the train to Munich Ost and the S-Bahn from there to the airport.