I am from an older generation that would find the use of the word babe to an adult odd.
BUT,the younger generation use it all the time,and so I cannot see what is the problem?
I think the problem lies more with you,and the fact that babe causes you to believe that there is still affection with her (on his part) when actually there isn't.
Life with a b/f who has an ex and child is never going to be easy,but don't make TOO much of a thing out of it or you may be an ex(soon) too!