I'm pretty sure you can't.

But there are things called "pancake boxes" that are basically just really thin junction boxes that are the width of drywall. Remove the current box you have and replace it with a pancake box that is centered (and on top of the stud).

There should be 2" of slack in the wiring to pull the box and install enough blocking to get the box to where you need it to be, then refasten the box.

I just spoke to the technical support rep at the manufacturer.
He "could not recommend anything like that..."

Looks like I have more work to do installing a pancake box!

Note that a pancake box will only work if there is only a single cable coming into the box.

More than one cable will exceed the box fill.

And I think the wire size has to be 14 gauge. Any bigger and it will exceed box fill.

Basically, it's a really tight fit with a lot of restrictions. But it may work in a pinch.

Can I attach one side of the mounting strap to the junction box and the other side into a drywall anchor?

Yup. As long as the fixture will cover the box with the right sided screw in the left sided bracket hole. Wiring will be behind the plate, as long as it is not pinched anywhere and you have enough wire it will be fine. Better to run the small fixture wires into the junction box and have the wire nut connections inside the box.

I was just about to post, but Vith beat me to it. I agree this is fixable without major re-work. My only suggestion is to use a screw-in type drywall anchor rather than the standard, crappy push-in type. That way your fixture base will go in tight to the wall...and stay that way.

Really? That's allowed by code?

I just assumed the light would have to be properly mounted to the box. I guess I learned something today. Sorry for the bad advice! :)

greg-

There is more than one way to mount a light on a box; some of them are designed to be held in place with only one attachment point. I've installed really wide fixtures where the hidden mounting plate screwed directly to the wall. While it would be nice for the OP to be able to attach both bolts to the box, I don't know of any code language that says how a fixture must be attached. The main issues are that the connections are properly made and wire-nutted, are within the box, that the wires aren't pinched or stressed, and the box is fully covered. Doing this as Vith described should be completely safe. I doubt that any inspector would have a problem with it as long as the fixture was securely in place.

You didn't give bad advice...it was just overly conservative for this unique situation.

Thank you to everyone that has commented! Tons of helpful information. I really appreciate it!

Rwie, what is the name for that animal? Havent seen it before but looks like a good solution.

Fan box

40 bucks! Thanks for the name.

Found one realistic price on menards.

$40? That Home Depot site is showing $4.55 to me. Must be location dependent.

$4.55 at my HD store, too....and a few other random HD locations I checked 1000 or so miles away.

Weird, anyways... buy it for a normal price lol.

If this is on a partition wall, not a supporting wall, the errant stud is doing nothing but holding up drywall. It can still do this even if you hack into it and move the box.

Thanks for asking this; my builder did exactly the same thing. arrgh