All interfaces between two different metals (bi-metallic interface) generate a voltage difference in the pieces of metal forming the junction sit across a temperature gradient. But...no temperature gradient, no voltage difference. Therefore, you can introduce any bimetal interfaces that you want as long as those pieces of metal do not sit across a temperature gradient, or a gradient small enough so that the voltage generated is negligible (either by having a zero or very low temperature gradient, or pieces of metal so small relative to the gradient that the temperature at both ends is approximately the same).
This means that:
The metal does not matter for the switch as long as the contacts are in a very small space so the temperature gradient is approximately the same everywhere around the contacts (which means the contacts have to be small relative to the size of the gradient which tends to mean they should not be sitting near heat sources where the gradient is highest).
If you are extending thermocouple readings across a temperature gradient you have to use thermocouple wire until the gradient has died down. Rephrased, this means that if you are using extension wires of a material different than the thermocouple, then you must ensure the temperature at both ends of the "segments of wire made of a different material" are at the same temperature, or else they will introduce their own voltage difference at the bimetallic junctions and distort the thermocouple junction reading.
So in all probability, the answer to your question is NO, the switch contact material does not matter since it's probably sitting far away from the heat source so the temperature gradient is very small, and the the contacts are probably very small reducing the temperature gradient found across the contact. You should use enough thermocouple wire to get away from the region of highest temperature gradient before you start introducing bi-metallic junctions.