This is plasterboard or something with a void about an inch back
This is the wall plug
The hole here is actually so big that the wall plug would fall through into the void if I put it back
I can't move the radiator up or down or left or right because it is fixed and even if I knew how to disconnect it I wouldn't want to spill black water everywhere and ruin the carpet, so I am thinking I should use the same holes but get a bigger plug
What about the plastic ones for plasterboard that sort of butterfly out when you fix them in?
I’m a big fan of this style of hollow wall anchor: https://toggler.com/products/snaptoggle-heavy-duty-toggle-bolts-zinc-plated-carbon-steel-channels Like the butterfly you describe, but the non-structural plastic bit keeps it in the wall while you get a bolt into it. The steel toggle is stronger than the folding type.
Even with the larger anchor however, there’s some chance that the wallboard is damaged, either as a result of the anchor failure, or as a cause of it. If it doesn’t feel sound, I like dem bosain’s idea of fastening a 1x4 to studs, then mounting the radiator the 1x4. What do you call those outside of burgerland? 20x95?