The definition of process would seem to say that my process is part of the larger company’s process because they are interconnected.  Why can’t the larger company just include my process in its RMP?

Your process is not part of the larger company’s stationary source because it does not meet the statutory criteria for stationary sources.  Although the process may be part of the same industrial group and is at the same location, it is not under control of the same person.  Therefore, the process is a separate stationary source and must have a separate RMP.