Oh Jim, how great would it be though if it did and the process inventor - or discoverer:-) would make a mint - have you already picked your thesis topic??
I did a google search for "heat melts fat cells" and it's been done already. The reason it doesn't work with a radiator is, I think, because it's not hot enough and anyway your skin is a fairly good insulator. But I'm honestly out of my depth on this one.
Re Thesis topic: Not yet -- well, sort of. Currently working on Higgs boson stuff, likely to switch to Beauty physics next year (this is actually a thing).
I guess it's a temperature thing. To melt the fat on a piece of steak you need it in a frying pan. A radiator just won't do it.
On the other hand, if you could sit in a frying pan.....................
The fat in you body is already molten at body temperature. The reason it stays where it is, is because it is contained in fat cells which are fixed in place. If it wasn't liquid it would be difficult to metabolise when you need a bit more energy ie. when you have run out of glycogen.
Blimey....amputation is a bit extreme.....Why not stand on your head while resting your bottom on the radiator and then have your ears syringed quite often?