The first ascent follows the easier (T3) but longer Landtalsteig to Wasseralm. You could save about 2km using the shortcut via Röthsteig (T4). Both routes are partially slippery and exposed. Be very careful. The descent is T4 too (IMO).

Only go on a day without risk of rain and preferably without rain in the previous days. The path is often slippery.
Approximate movement durations with a fast group: 2-2,5h Salet to Wasseralm. 1h Wasseralm to Halsköpfl. 45min Halsköpf to Grünsee. 2h Grünsee to Salet.

The first boat departs at 8:30 (arrival 9:25 at Salet) and the last return departs at 17:40, i.e about 8 hours left for the hike. https://www.seenschifffahrt.de/de/koenigssee/

If you need more than 3h to Wasseralm or are unsure, turn around at Wasseralm and skip Grünsee.

optional extension to Funtensee (coldest lake of Germany, 6km and 300m gain extra) - only for fast hikers, see https://www.hiking-buddies.com/routes/routes_list/1667/