The simpler answer is that with CPL/IR and "only" CPL writtens, you can do anything in an aeroplane certified for single pilot operations. You need the APTL writtens to fly RHS in multi-crew aeroplanes, and the full ATPL to fly multi-crew LHS.
I believe that aeroplanes certified single pilot but operated within company procedures 2-pilot still only need CPL/IR.
As has already been said, FI only needs CPL writtens and not the CPL skill test. It doesn't need IR either, unless you plan to also become an IRI.