I haven't chimed in because I'm not really 100% sure, but here's my thought. A K-Type is a thermocouple. It has two distinct types of metal that, when heated or cooled, alter the resistance in the thermocouple. They work because you calibrate them to your system based on a set length, gauge, and type of cable. If you run them through a switch, now you're altering the type and length of cable between the thermocouples, which results in the system having no idea what it's measuring and the thermocouples not working. If you want to have a switch that will allow you to go from one temp probe to another, my suggestion would be to get 2 RTD probes and a switch with 3NO and 3NC terminal blocks. Run one RTD to the NOs, run the other to the NCs, and make sure you keep the wires in the same order from both RTDs. Then jumper the terminal blocks on the non-RTD side, and run 1 wire for each pair of blocks to your PID. Confusing the way I wrote it, I know, but it should allow you to switch between the two.
Or... save yourself the headache, get a plug end on the panel like most people on here use, and just physically switch out the RTDs when you want to swap temperatures.
Short story - probably won't work with K type... (or, I'm an idiot).