What height 2 piece mount is required to clear the front sight post on an AR, but still allow a decent cheek weld on a standard carbine stock? 3x9x40, 1” tube.

1.5" from rail to centerline of the optic is kind of the 'standard' for optic height on AR-15s.

It won't 'clear' the front sight, but you won't see it with the magnification on that scope.

There are some higher mounts, up to 1.93 and even 2.03, but those are usually for a very heads up shooting position and to clear PEQ / IR lasers.

And for an AR-15, without a monolithic front rail / receiver, I would highly recommend a 1-piece mount with eye relief built in, otherwise the scope will be too far back, or you will have to place the front ring on the handguard rail.

Not seeing a shadow from the front sight tower and good cheek weld are opposite ideals in my experience.
I use a one piece cantilever mount to get the optic far enough forward for eye relief.
The shadow is no biggie in the 3x10 I have on my varmint carbine.

Not exactly same situation but I have an ACOG mounted on my AR with a standard front sight (Used on duty for years). Front sight isn't even noticeable. I would mount it so it's comfortable. If you raise scope and need to add a riser you can run into problems with the charging handle hitting riser if using collapsible stocks.

Agreed. With a magnified optic you won't see the front sight post, so go with the mount that's comfortable for you in both height and eye relief.

Yep. Fixed front sight posts are only visible with unmagnified optics. There may be a little bit of noticeable ghosting with a 1.5x or 2x optic. But you won't even notice it with a 3-9x.

You're focusing past the front sight post. Like watching the action on a ballfield while standing directly behind a chain link fence.