Where is snow in late August?

Late August is probably the most unlikely time in the whole year to find snow. I mean, think about it - August is the end of summer, a hot time, with no snow for months anywhere in Austria. The only place you might see snow is very high up on a glacier, such as Hintertux. Here’s a link - https://www.hintertuxergletscher.at/en/home/

I think Kaprun might be a good bet.

As far as I know, there are only two places in Austria where is not only always snow in the summer but also a touristic infrastructure for visitors who do not want to go skiing or mountain climbing.

(1) Kitzsteinhorn
https://www.kitzsteinhorn.at/en

(2) Hintertux Glacier (as Emily already noted above)

Global climate change is impacting skiing year round but try this:

Dachstein Glacier Wikipedia:
Location Ramsau am Dachstein, Austria
Top elevation 2,700 metres (8,900 ft)Base elevation 2,057 metres (6,749 ft)
Website https://www.derdachstein.at/en
Dachstein Glacier is a glacier located above the town of Ramsau am Dachstein in Austria

https://www.derdachstein.at/en/dachstein-glacier-world/ski-and-snow-active/cross-country-skiing

From July to November the world's elite has the opportunity to enjoy
ideal preparations for upcoming competitions on 18 kilometres of
diagonal and skating trails.

Depending on where you will be, the Zugspitze in Germany may fulfill your wish, reached from the Austrian side at Ehrwald. No idea what the conditions are like this year.

Was around Zugspitze three weeks ago. No snow. Sorry to report this.

I have seen snow on Grossvenediger in mid-August several times. If you are close to Kitzbuhel, drive south toward Mittersill and you can't miss it.

Real time (short time delay) Dachstein Glacier webcam:
https://dachstein.panomax.com/
Helps to view early morning North American time as the night itme webcam view doesn't display the snow very well. (7:50 AM PDT morning = 16:50 Central Europe Summertime afternoon).

Thanks Everyone. Will have one day in Berchtesgaden, and 2 days in Zell Aam See, I hope I will find at least a little snow in some of the places in Kaprun that were mentioned by you