Fussen to Berchtesgaden drive

Perhaps Mittenwald? In Austria maybe The Zillertal?

Go along the German Alpine Road/ Deutsche Alpenstraße

Great info on the website for what you are looking for

For ex stay a night at Gasthof Keindl in Niederaudorf on the way
Near(5min) Oberaudorf on the Alpine Road ca.2 hrs from Berchtesgaden
We stay there for a night when going to Südtirol/AltoAdige or Gardasee/Lago di Garda

or when you like Hiking in nice weather

Stopover at Walchensee little Village on Lake Walchensee
here Pension Edeltraud
In the village is the Herzogstandbahn Cable Car Go up and do the Hike to the summit
from the Gasthof/Restaurant on the mountain (1,5km)
One of the best views in Bavaria

Both villages split the way from Füssen to Walchensee (90km) to Niederaudorf (100km) to Berchtesgaden (120)
in 2hrs scenic drives along the German Alpine Road

For me I would head south into Austria passing the Zugspitze, over the Fernpaß and continue towards Innsbruck along the Inn River. You could consider an overnight in Alpbach, Achensee or Zillertal. Continue the next day passing the Wilden Kaiser through St. Johann in Tirol and back into Germany to Berchtesgaden. Stay on the secondary roads. There is no toll or Vignette requirement when you do not drive on the Austrian Autobahns. Also, with noise walls and tunnels on the Autobahns you will see more on the secondary roads and drive through many small towns. Be aware of speed cameras along the way. Relax and take your time. June is a great time in the Austrian Alps with the wildflowers. One of my favorite things is driving through the Alps with the top down.

Please check with your rental company for driving in Austria, and Austria for licenses and Visa requirement from where you are coming from.