You should use a ceiling fan rated box. I’ve used this one with 100% success:

https://m.youtube.com/watch?v=gz5HIFBb-e8

Would this work if the stud runs directly adjacent to the hole in the ceiling? Or you think I should move the hole?

The box slides all the way across. It will work for you.

Home Depot sells a few varieties for either floating between joists or butted against a joist. Also make sure you get a retrofit box if you don't have access to replace it from above.

I’ve been searching there - is something like this what you mean? Looks like this could work.

Link

That would be perfect. This type would also work.

Your old box is probably nailed into the ceiling joist, like this box, so expect that it might be a bit difficult to remove. I've sometimes had to use an oscillating tool to chop it to bits in order to remove them before.

Appreciate the reply. I figured so, I’ll have to see what I can do to get access to those nails. Thanks again!

Yes, that one or the other one that was suggested would work.