With a 7h45m layover from 23:45 to 07:30, you actually do have time to step out on your own, as long as you keep it simple.
Doha is very doable by taxi at night. Immigration is usually quick, and taxis are easy to get. The city is safe and well lit, even late.
You won’t get “open attractions” at that hour, but you can still see a lot:
Msheireb / old Doha area for a walk
Katara from the outside
The Pearl for a quick drive and photos
West Bay skyline at night
Souq Waqif is hit or miss late at night, but sometimes cafés stay open into the early morning, especially on weekends.
I’d plan to head back to the airport around 4:30 to 5am to be safe. That still gives you a few hours out without feeling rushed.
Just don’t try to do too much. Pick one or two spots, enjoy the night views, and head back.
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