I don't think the bartender is a major player. He just provided an meeting location for the Rosato/pentiangeli meeting.
I think Carmine drew out his gun when the cop began to walk in the backroom to investigate what was on the floor. So I don't think his intentions was to shoot the bartender! Now, if the Rosatos really wanted Frankie dead, why did they kill the relatively unimportant beat cop, along with Frankie, and dump their bodies in another location? The mob killed a NYC Captian, what would a small time beat cop means to them?
Again, I am still unconvinced that Michael is so concerned about legitimacy, or his Public relations rating. In GF I Michael did want the Corleones to be legitimate "all the way", as he said to Tom. But, the ongoing theme of GF II is how deeply corrupt Micheal became.