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.

In fact, they already make the stuff: https://nano3dsystems.com/product/copper-nickel-conductive-fabric/

They make Stainless too - but it only consists of 30% conductive material. https://nano3dsystems.com/product/stainless-steel-conductive-fabrics/