Few people get along with all of their neighbors. Explain your situation factually, and when they talk to the neighbor they will know the truth. BI's are professional interviewers.
Also, I don't really know any of my other neighbors too well. How could they use them to reference my character in these circumstances
Easily. Your neighbors will know if you have loud, obnoxious parties. They will know if you drive up and down the street like a maniac, etc.
"Speed is fine, but accuracy is final" --Bill Jordan
Remember those who died, remember those who killed them.