They didnt need to invade US, they simply needed to keep them from europe. Sooner or later the public wants peace. I dont think the German people would had surrended even if US would drop A-bombs since they didnt even surrender with soviet troops rampaging in Berlin with most of germanies cities in rubble. And also how many A-bombs would the US public tolarate being dropped in Europe? Many ordinary people in USA didnt want US to get involved in the war at all.
Well, if Japan with their absolutely fanatical kamikaze attacks and banzai chargers surrendered after getting nuked,germany would of too. Also, you don't need to surrender , nukes would simply screw up all logistical/infrastucture/command centers and german high command would be next to useless.
You also seem to underestimate the power of a nuke. A few nukes would kill more than all german civilian losses during the entire war.
As for USA public don't worry, they had this thing called propoganda too.