My 3 year old is also an early riser. We put her down at 7:30 and she usually plays in bed until 8:30 before falling asleep. She does not sleep during her “nap”, though she will usually lie down for the first 10-20 minutes, so that’s something.
I think she needs more sleep as she clearly show signs of being tired by late afternoon. But we can’t force her to fall asleep, just try to create opportunities for it. We sometimes plan outings in the car or stroller on weekend afternoons, which we know will almost always lead to her passing out, just so she gets a little caught up on her sleep.
I think this is a tough age because there are lots of changes in brain development and it makes it hard for them to relax. My approach is to maintain her schedule best we can and hope she settles a bit in the next year or so. Or she’s just doesn’t need much sleep and she gets used to playing on her own in her room every morning until we get up.