I'm booking a trip that will require two separate tickets and I'll be connecting in FRA on my way home to the US.

Background:
I'll be coming from Slovenia where I will have spent 9 days
Flying to FRA on LH Cityline, likely arriving midday (12pm)
Flying to JFK on Singapore early the next morning (8am) (booked Suites on points which is why this is two itineraries)
Double vax, booster, and plan to get tested the morning I leave Slovenia (for return to US)
I should have only carryons unless LH makes me check something

In short, I'm trying to figure out transit requirements and options:
I don't believe I can "enter" Germany since I don't have a "need" to be there.
But can I stay in sterile transit when coming from a Schengen country and departing for the US?
Does having separate tickets complicate things (either departing LJU or arriving FRA)?
Where do I check-in with Singapore if I don't leave transit? I assume they have a transit desk.
Can I stay at an airport hotel? Would My Cloud Hotel work (it looks like you can't check out until 8am? I'd need to be out by 6.) Or just plan to sleep in the terminal?

Anything else that I'm missing? Thanks for any help. I always appreciate the expertise on FT.