Welcome Caption ! I got a 3-way who has seen a lot in his life. Among other things, it had a similar problem with the mode switch. I fixed it the same way you did by soldering the plate to the shaft a little. The selector began to work, but there are still a couple of problems, one of which I have already solved, but there is still a problem with the operation of the left channel.
Could you share the schematic that you have? Otherwise, I will have to sketch and understand the device for a very long time. The layout of the crossover is very compact, and to be honest, it's not entirely clear to me why it was needed in such a large case.
I really hope for your help, thank you!
I apologize for the spelling syllable, I am writing through a translator.