If things with the Bears are still horrific in year 4, Caleb could tell the Bears if you pick up my fifth year option just know I won't be playing for this organization after my 5th year. I'll enter free agency and go somewhere else. The Bears would get zero compensation in this scenario. And if Caleb, by this time, had developed into a top tier QB he would have significant value on the open market. The Bears would trade him in year 4 so they get something in return before he hits the open market as an unrestricted free agent.
Players don't have to sign the franchise tag. That creates a lot of problems for them but Caleb does have the option to not sign it.
I admit that my scenario is far fetched.
Personally, I believe the specter of Caleb leaving before he signs a second contract is a good thing. It might be the only thing that could force this organization and its leadership to get serious about doing things the right way, stop cutting corners and legitimately try to build a winner. If you do things the right way then Caleb never leaves Chicago and we are in the SB window for many years.