Cutting the plug off and resoldering the wires may add a resistance to the probe and will change it's temperature. By how much, not sure. Probably not much though, just make sure it's a good solder connection.
The resistance at 200F is something around 7300 ohms which is still pretty high compared to contact resistance. Room temp is like 100kohm. I think the reason Thermoworks suggests to only use one extender is because the longer the wire the more noise it is going to pick up because they aren't shielded