Maybe I should talk to my boss and try to convince him that people at
C will learn that I left anyway?
I think that is your best strategy. I suggest pointing out to your Company A boss that Company C people are going to see the update to your LinkedIn profile after you leave A. Claiming you are on sick leave is going to look really silly when your ...
I was in a similar situation once - I was employed by a parent company to work at a client company, and when I resigned my manager asked to say nothing, so the company I was body-shopped out to would stay happy. I did what my employer asked.
I stayed in contact with some of my ex colleagues from the client company, and when they found out that I didn't work ...
Short answer: Don't lie.
Your professional reputation has value
If you participate in this fraud, and yes, it is fraud, by "Company A", you will be in the very least tarnishing yourself as being a liar, untrustworthy, and unprofessional.
You could expose yourself to legal troubles
IANAL and you should check with one as depending on the ...