Check/bet (non)backdoor draw ?

[minSim]

This is just an example hand, fill in any hand you think of checking behind on a 2tone flop. If you need more info, please ask.

So let's say we're in a 100NL, 6max game, 100bbs.

Hero raises LP, get's called from the blinds by an unknown .
(villain is no extreme, will likely lead turn with a pretty wide range (that is standard isn't it?)).

Flop comes like 378, villain checks.

Let's say I'm thinking of checking behind i.e. A6,
should I be more inclined to check back A6 or A6 ?



3 thoughts of my own and I can't choose;
- check behind the backdoor draw, you'll be able to profitably call or raise more turns
- bet the backdoor draw because you can extra barrel the spade turns (and it basically just has a little more equity)
- it matters little / check both or bet both

[meeloche]

Well villain dependent I'd likely cbet both however I'm more likely to cbet the spade one cause I can barrel.

[bode]

Well villain dependent I'd likely cbet both however I'm more likely to cbet the spade one cause I can barrel.

[minSim]

I'm not sure if you picked this up or not, but it's not about this particular hand and board. Just take any 2tone board you're thinking of checking behind in which you think the decision is close.

Could you name something that's villain dependant that's reasonably for a standard Micro/SSNL player that makes your decision go one way or the other?
(which will make you choose to cbet the backdoor draw and the non-backdoor not, or the other way around)