Hi Steven,
You answered your question more or less yourself already.
As long as you can frame your experience in a way that PMI accepts, the rep is correct; you should go straight for PMP; doing CAPM first would be a waste of time in your case.
To frame your experience correctly, you should use the PMP Exam Content Outline and the vocabulary from the ECO; this ensures alignment with the PMI requirements and their way of thinking.
I am pretty sure that would be easy in your case, given the vast amount of experience you already have in leading teams and managing tasks.
Feel free to reach out (on LinkedIn, for instance) for any further guidance or support with your application. I would be more than happy to help.
BR,
Markus