One data point:

I flew EWR-FRA-EDI this summer (United and LH) and I had to clear passport control into the EU and then another check to leave Schengen as there was no way to get from where I landed with United to where my EDI flight departed without going into the EU. I did NOT have to go through security again.

If you experience the same thing, it would probably make sense to get to your gate area first, then use the closest lounge.

Added information you didn't ask for: I ended up misconnecting (we were late arriving) so I had a much longer layover in FRA than anticipated. If that happens to you as well you can use the lounge staff to finalize your rebooking and get paper boarding passes and skip the long customer service lines. I was automatically informed of my new flight on my Lufthansa app, so it might make sense to download that as well.