be much harder to shoot potential enemies when they literally are your friends
If this were the case, the US wouldn't be notorious for high numbers of police shootings. It's far more likely that if there is some kind of US civil war it would be race-based, like the previous civil war and the Indian wars.
You're assuming the police and the military attract the same sort of people, keep them under the same sort of discipline (or lack thereof for the police), etc. That's not my impression at all.
In fact, there tend to be two different types of militaries, the expeditionary type, which is ours, and the internally focused type, which our increasingly paramilitarized police seem to be becoming. The latter suck when they're asked to fight a "real war", and one advantage we in the US have right now is that their numbers are quite small compared to the size of the nation or our formal military, something less than a million total.
If this were the case, the US wouldn't be notorious for high numbers of police shootings. It's far more likely that if there is some kind of US civil war it would be race-based, like the previous civil war and the Indian wars.