I sometimes do sleep in a tent outside in the garden (Not out of heat reasons yet, but because of fun events with the kids).
I wouldn't worry about being woken up by the sun, as what will actually wake you is not the sun, but the relentlessly enthusiastic birds chirping away at the top of their lungs almost an hour before sunrise already ;-)

I did this when I was visiting Mexico. Probably some of the best sleep I've had in my life. Balcony was on the North side of the building. Even though it was hot outside, the extra airflow through the hammock kept me pretty cool.

You might want to try a sleeping mask