If the lead wire is copper, there must be something else compensating somewhere else and I agree, a switch would not change anything. Current capacity is really a non issue here as we are talking about millivolts. I would see good quality contacts as a more important issue.
The Type K lead wire extension wires that I have dealt with were also of Chromel and Alumel. That is why the color code matching is so important on the leads between the TC and the electronics. I assumed that the leads I got from Isspro were matched alloys as well but I will admit that I didn't check them to see if they were copper.
If I remember correctly, to see which is Chromel and which is Alumel, one is magnetic and the other isn't. If my wires weren't buried in loom or under the dash I would whip out a strong magnet and check it to see if one of those wires was magnetic.
Ken