You lucky, lucky thing !
I did a four day walk in that area a few years ago and it was beautiful. Hard work, but beautiful.
We stayed in the AlpenVerein (Alpine club) huts. If you do that then it is worth joining the UK branch of the AlpenVerein aacuk.org.uk. It gets you discounts in the huts and also importantly it includes insurance.
Anyway, I digress.
The route we took was to start at Konigsee. Take the boat on the lake to St. Bartoloma chapel. From there we walked up to Karlingerhaus (1630m).
Day 2 we walked across the Steinernes Meer to RiemanHaus (2177m).
Day 3 from Riemann Haus to Wimbachgries Hutte (1327m)
Day 4 from Wimbachgrieshutte to Ramsau.
If you are experienced walkers and fancy a bit of an adventure then I can thoroughly recommend that. If you fancy something a bit gentler then the boat ride on Konigsee to St Bartholoma, followed by a little stroll around there would be good. But to be honest it's all lovely.
p.s. We went in late May when there was still a covering of snow on the Steinernees Meer. I haven't actually seen it in the late summer. I imagine it could be very barren up there.