Stainless is an insulator of heat. In fact, it's used in 3D printers to avoid heat traveling up the path where the filament is fed in from.
Copper with a silver or rhodium plating would be a good choice otherwise. Copper has a higher thermal conductivity than aluminum, and silver does as well, but is also hypoallergenic.
I wouldn't call it an insulator but it is a poor conductor of heat compared to aluminum. I think in this situation either one would be capable of dissipating heat better than just about any blanket.
Right, like I said a poor thermal conductivity but as far as blankets go, you won't find anyone attempting to get warm with chainmail because it would unfortunately result in an increased loss of body heat. In so far as blankets go, not an insulator.
Also worth mentioning I was trying to stay in the real of possibilities and this case stainless wire is readily available in many forms for welding, and would resist oxidation better than copper. Obviously we can just name the best conductors of heat.
Copper wouldn't be difficult to plate in silver; you can buy plating solutions pretty easily, enough to plate a blanket easily. Copper is also incredibly easy to get, and not only has the highest thermal conductivity but also the highest weight as well. (of malleable metals at least; tungsten beats it out weight wise, but couldn't be easily formed)
Stainless also work hardens much faster than copper. Much bending at all and it starts to fracture. It wouldn't last long as a blanket - though there are ways around that, such as how you make the blanket; chainmail, etc.
Copper can be thin enough, and malleable enough to be woven like a standard fabric as well, resulting in a 'softer' feeling.