よし 後ろ盾を得たところで さっさく調べちまおう
"後ろ盾を得た" is a way of saying that means "To obtain one support". The character proposed an hypothesis about the situation and his partner agreed, which prompted this.
My reading of the sentence is akin to "Well then, now that I have obtained your support/agreement, let's investigate"
However, why ところで here used? I don't think the "even if" meaning makes sense?