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’d suggest looking at hollow wall anchors (also called umbrellas fixings) which are similar in design to the ones you’re describing but heavier duty (steel) and designed to hold the weight of a radiator, though they can be difficult to use if you’re just using a screwdriver (in my limited experience.) Also a new product called Gripit which I have no experience with but apparently are designed for this sort of thing. Seems quite a bit easier to use.