Suppose we have some suited hand - say AQs - raise pre, and the flop comes 972r. Of these from best to worst, would it be best to cbet when we have the suit of the 9, the 7, or the 2? If I want some method to use if I decide to cbet a certain hand with a BDFD and no pair only some percentage of the time, is it always the same -- Like high card to low card or low card to high card... Or does it depend on specific flop texture (like is 972 different than say 923 in terms of BDFD "priority"). Also, does this priority change if we're facing multiple villains vs going heads up?

I'm sure in general game dynamics are a much more important factor, but it would be nice to have some sort of system at times.

When ranges are narrower (e.g. in 3-bet pots), I think the blocker/removal effects can slightly affect the optimal frequencies for betting with BDFDs, such that you'll sometimes be better off betting the BDFDs that match the highest card on the flop (or conversely checking back that combo, but betting the other BDFDs at a higher frequency) on some flop textures, but I think the effect is insignificant when ranges are wider, such as in single raised pots. When ranges are wider, I think the ranks of your cards are usually more important than their suits.
I'm generalising massively though. I'm sure if you studied a bunch of Pio solutions you could find some quirky spots where on 972r you might bet some Q8s that match the suit of the 9, but some Axs that match the suit of the 2, or even some Axs in the suit that has no BDFD at all (while you check back the BDNFDs). There are usually so many mixed strategies that it's almost impossible to spot "patterns" in things like this.

Interesting... Yeah I was hoping that there were some definite/simple patterns or rules for this, but I guess it makes sense that there aren't.

Agree it's extremely insignificant but from the arm chair it seems like you'd rather the 9 be of the same suit, because you'd rather your opponent not be blocking your flush as well as having the 9. The odds of your opponent blocking the back-door flush draw should be reduced. Similarly if it were a flush draw rather than a back-door flush draw, you'd rather both the 9 and 7 be of the same suit, because if they had a pair and a flush draw at least it is only a pair of 2's.

You would probably be more likely to be paid off when you hit a BFD using the bottom card because your opponent is more likely to continue with top or middle pair than bottom pair, so a BFD using the bottom card on the flop is more likely to lead to a situation where you have a flush vs trips when your flush includes a turn or river card that pairs the board.

It is possible that the effect of this is insignificant.

Solvers prioritize high card removal and backdoor straights. So in your example of 972r some of the highest frequency BDFDs being cbet are A6, A8, A3, A4, A5.

From my experience, mainly in single raised pots, it doesn't particularly matter what suit you hold, they're all bet pretty much equally. The type of hand you have makes way more of a difference than the suit.