The Jennerbahn (mountain lift, renovated in 2019) is in the same parking area you park and walk towards the Konigsee boats. Others can speak to the public transportation, but I think you can do both in one day. More if you are adventurous. Watch the weather. On a good day you can get both bright sun and pouring rain. Come prepared. If the weather is socked-in, then have a plan B (Berchtesgaden, St. Sebastian Ramsau bei Berchtesgaden?).
My opinion is the Jennerbahn and Konigsee is time much better spent than the Kehlsteinhaus. Everything should be open in June (Alm etc.).
In my limited travels, the Konigsee is one of the most beautiful places I have been. Been there a half dozen times or so now and have taken Family from the US there, including Mom and Dad. Multiple hikes in my younger days. I'll go again someday, even if just to sit there and soak it all in.
https://www.jennerbahn.de/
https://www.jennerbahn.de/webcam-jenner/
https://www.foto-webcam.eu/webcam/funtenseetauern/