Typically, host nations will turn US on US crimes back to the military even if the crime occurred off post. They would not want to bother German taxpayers with the costs of a trying an American service member for the murder of another American service member. Had the victim been German, I can pretty much assure you, the Germans would have retained the right to prosecute. While I agree this has all the makings of a homosexual love spat gone wrong, I don't think there was any lavender mafia pressure on the Germans to turn the suspect back to the US. There will be ample pressure from that crowd however, to keep the trial details out of the press if things are as they appear on the surface.
It appears that the German prosecutor was a little bit reluctant to give this one over to the Americans. “According to Card, the German prosecutor’s office is reluctant to fully release the case,,,,” with the American response of not accepting the German evidence and not bothering to even arrest the guy until publicity started to build.