Will completing this quest put me at odds with the NCR?

pretty much yeah

Ok, then I'll just drop it...I got past "check on the wounded" part and thought that may be the case.

It didn't when I did it.

Don't think it does, because the Ambassador guy tells you that you might have to do tasks for them and be undercover.

Didn't mess up my NCR rep. Long as you don't kill too many NCR personal what rep drop there is can be countered by playing with Cook-Cook and his pals.