Here's my experience report update.

They acknowledged I had a business class BP but said because the next leg was economy I'd have to pay a 29euro fee to enter the business lounge with a Lufthansa BP in the G gates area.

Worth it in this case because it was about a 5+hr layover and they do have (two) showers in the lounge as well as hot foods and an open bar and free wifi. You'll need to find the attendant to get into the showers though. They have a sign up book, but they don't call your name or anything.

Regards,
-Bouncer-