Repair hole near stud

We've just moved into a new house and one of the radiators was tilted. I've taken it off and it's because one of the brackets has come out of the wall, and is being held solely by resting on the plasterboard below it. Ordinarily my plan would be to square off the hole, put a piece of wood behind it then put a replacement piece of plasterboard onto that.

My issue is that one side of the hole is fairly close to the stud (roughly red line). I'm not convinced that there's enough room for me to get a piece of wood in sufficiently that I could secure it on that side without being too close to the original damaged plaster. I did consider putting the bit of wood in vertically, but then I encounter the same issue with the proximity of the screw to the damaged plaster below it.

Assuming that all makes sense can anyone suggest the best way to go about fixing this please?

Thanks

Go watch.... On the trowel... YouTube. Kirk shows how you repair plasterboard.

Cut board big enough to cover 2 vertical studs.
Place on wall and draw around.
Cut wall to lines.
Put some wood in for rad brackets.
Fit plasterboard
Make good and sand smooth
Paint
Screw rad brackets on
Hang rad

Perhaps I'm misunderstanding, but a board big enough to cover 2 vertical studs would be 400mm wide - seems excessive when the hole is only 40mm or so wide to start with? Also when you say 'put some wood in for rad brackets' what should I be attaching it to? I'm assuming that it would be screwed into the existing good plasterbard, however then I go back to my original query of not having enough good plasterboard available below or to the right of the hole to use.

Apologies if they're daft questions!

Hi
Quick look on YouTube I couldn't find anything that matches your problem.
I've fixed rad brackets many times.

You could cut out rectangle ready for small plasterboard. Sister some CLS to stud £4 with 75mm drywall screws off Amazon.
Buy handy board although 9mm thick but that don't matter. Pack out with something. Handy board is a small peace of plasterboard from DIY.
Fit board with 32mm drywall screws off Amazon.
Fill
Sand paint
Fit bracket to sistered CLS timber.

Or

Find vertical stud to the left and right of hole.
Mark line in the centre and cut out.
Now fit horizontal timber so brackets can fit to.
Now fit plasterboard to opening.
Fill
Sand
Paint
Fit bracket to horizontal timbers through plasterboard.

Board is £9.50
Cls timber £4
Box of filler
32mm drywall screws £3.23 Amazon

Watch YouTube for fitting and filling demos