There were two switches used with the SII Britax column lock like you have. There was a phenolic based one and a plastic based on. I know the plastic base switch is notorious for getting hot like you describe. I don’t know about the phenolic one. The contacts inside the switch get corroded and oxidized which creates resistance and the result is heat. The only thing you can do is disassemble the switch and file and lightly sand all of the contact points. Also the spades on the back need to be very clean. Check to see if there is ANY resistance between the rivets and the contacts. Any resistance must be corrected.
This is a picture of the back of the plastic base switch. The Lucas part number is 39679A
This is a picture of the back of the phenolic base switch. The Lucas part number is 39679B.
To disassemble the switch you have to VERY carefully bend back the three tabs that hold the back on. VERY slowly remove the back noting where everything is located. There is a ball in there that facilitates the spring-back lock out. Be sure you assemble the switch with ball and spring tabs in the correct positions. Here are some pictures of what the inside looks like:
I used some epoxy to assist the tabs when I re-assembled mine because the springs tend to try to keep it apart. With the un-avoadible damage you will do to the tabs, it really helps keep the switch together.
The wires on the back of your switch appear to be heat damaged. That should be fixed as well.
The best thing is a new switch but these have been NLS for the SII for many years. As an alternative the SIII switch will fit your lock and work fine. It is still available from the usuals. See https://www.sngbarratt.com/us/#!/English/parts/9a8a9460-c7f6-4800-a290-434506447073 The difference is that the SIII switch has no spades and the wires riveted onto the plastic base. The wires go to a connector so you will need to use your imagination to connect the wires from this switch to your harness.