I’ve read horror stories about parking in Hallstatt. We plan on going to Berchtesgaden for Eagles Nest and Documentation Center, then Hallstatt for lunch and Salt Mine. Should we take a bus rather than our rental car? We are staying in Salzburg.
I stayed in Salzburg and took the bus to Berchtesgaden, then changed to another bus that takes you to the Eagles Nest. I’m sorry I can’t help with driving to Hallstatt.
Are you planning on visiting Berchtesgaden and Hallstatt in the same day? If so, I would say you will be shorting both those sights.
Although you can certainly drive to each place in a day from Salzburg, visiting both in one day won't leave you much time to really enjoy either place. Your total drive time will take 3.5 hours round trip at least. I would especially recommend the village of Hallstatt, which is best enjoyed in the late afternoon or very early morning, be an overnight stay. During the heart of the day at lunch time all the bus tourists are there in full force. You can access Hallstatt by train or bus and leave your car in Salzburg. There are bus tours to Berchtesgaden so you really don't have to drive there as well. Good luck and enjoy Salzburg.
If any of you are interested in Formula 1 or historic aircraft, don't miss Red Bull's Hangar 7 at Salzburg airport
Doing Berchtesgaden and Hallstatt in one day does not make much sense, not to talk about that Hallstatt is best reached by public transport, preferably as a day trip from Salzburg.
For going to Hallstatt see here:
https://community.ricksteves.com/travel-forum/austria/how-difficult-is-it-to-find-parking-in-hallstatt-in-early-june#2466828