Help with quality in printing recess

[ptchan5]

Not sure if this is the right place to post this, and I'm new to 3D printing and the forum so please go easy.

I am designing a part that basically has a recess on the bottom. Think of a standard tea cup type shape that normally has just a rim that touches the ground and the center section is recessed into the bottom of the cup. When I print this I do get a recess, but the recess is not that pretty because as you would expect, the roof of the recess has no support. So when the printer goes to string across the opening it lays down long unsupported threads. My problem is not so much the sagging that you'd expect to happen, but the roughness of the unsupported threads.

The design can't be flipped 180 degrees because I have something sitting on top of that "roof".

How does one design this so that the "roof" section of the recess is smooth or at least nice enough to be considered a finished part instead of poorly printed piece. I tried little support extrusion pillars with the thought that the printer would tie the cross threads to them, but that didn't happen. Came out exactly the same. If I make something too solid it will be difficult to remove.

How would you experts do this?

Thanks in advance,
Newbie (Peter)

Attached .stl file for review.