They haven't asked to see it.
and halt the never-ending lawsuits
It wouldn't halt them. Birthers would claim it's not enough. They'd also double down on the argument about his father being a British Subject, or him supposedly losing his citizenship while in Indonesia.
At least half of the lawsuits filed to date are unrelated to the birth certificate.
and bad publicity?
The publicity helps him. It makes his opposition look like unhinged lunatics.
What’s unhinged is Zero and minions spending habits shutting down any transparency dealing with these lawsuits fighting to avoid subpeona.