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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?
This (た)ところで has nothing to do with your sentence. ところ refers to a situation and で is a location marker, so the literal tranlation is "in a situation (where ～)", but "now that" is perfectly fine in this context.
たところで can also refer to a physical location, for example 門を出たところで待っています "I'm waiting outside the gate".
“〜たところで” can mean three things the way I see it.
The idiomatic “even if” or “even assuming”. This usage doesn't really combine with the volitional form and is usually followed by a negative sentence or at least a sentence that resigns to an unfortunate fact. “後ろ盾を得たところで、調べられない。” would be more befitting of this use, as in “I couldn't investigate it even if I could gain his support.”
It can also simply be used in the literal sense of “ところ” as in “place”, such as “あなたが行きたいと言ったところでデートする” to mean “We'll go on a date at the place you said you wanted to go to.”.
And then, the final sense, the one used here, which is simply the te-form of “たところだ” as in “just after doing”. “後ろ盾を得たところだ。” means “I have just gained your support.”, of course, like any other Japanese sentence this conclusive sentence can be put into the te-form and connected to another sentence, so it means “I have just gained your support, so let's investigate.” or indeed, more idiomatically. “Let's investigate now that I [just] gained your support.”
