Best way to temporarily relocate a radiator?

I think option 2 sounds easier, I can even do it without draining the system

Yes, not draining the system is a huge plus. You'll have to drain the radiator of course.

but (other than the weirdness of the valves not being next to the radiator) are there issues with that approach?

Yes, you have to avoid having high spots in the pipes because air bubbles will settle there and will be impossible to remove. The pipes must have a little bit of slope to allow air bubbles to rise into the radiator, then you can open the air trap on the radiator to let the air out. This means the new spot for the radiator should not be lower than the old spot, or the pipes will have slope in the wrong direction. If you use PEX pipe which is easier to work with, then it would be better for the new spot to be like 1" higher.

Or, are both these ideas stupid and I should do something else?

It's a bit of a kludge, but hey, it's your radiator, so why not.

One extra detail, just to make life interesting ... the radiator feeds are 10mm piping, but I would probably just use 15mm for the extension. Or is that a bad idea too?

No problem, but have a good look at what kind of fittings are used to connect the valves to the radiator. You will need compatible fittings, and probably more fittings to connect that to your pipe, which may decide what pipe you'll use if you want to minimize the number and/or total cost of fittings.

Also I'd really advise using PEX pipe or other type of pipe that is a bit flexible, because you won't drill the holes in the wall with absolute precision, so in the end the radiator may not be exactly where you want down to the millimeter, and if you end up with steel tube that is a few millimeters too short or doesn't end up exactly where you need it, well you can't bend it in place, so it's kind of a bummer. Worse, the weight of the radiator may end up resting on the pipe and not on the wall.