Thanks again to everyone that has suggested ideas. After a bunch of trial and error, I believe I figured out the problem, which in turn revealed the issue was something entirely different from what I originally thought.
In terms of composite output, I do believe either the din connector or the cable itself is faulty. The cable does not fit snugly into the port. In fact, it sort of just hangs in the port as opposed to snapping in flush.
With that being said, my intent all along had been to use the RGB cable I purchased that connects via the EXT port. I discovered the reason that too was not working had nothing to do with the console or cable itself and all to do with my setup.
What I'm doing is the following:
PC Engine --> RGB Cable -->
gscartsw Scart Switch --> Framemeister
It turns out there is an issue with the gscartsw switch properly receiving the video signal through the RGB cable. If I bypass the switch entirely, and just plug the cable directly into the Framemeister, I get a nice clear picture without issue.
After contacting fidde_se, who makes the cable, I learned it is a known issue and he is able to make an adjustment to my cable to make it work with the switch.
Hopefully my experience can be a good reference point for anyone planning to go down a similar route in future.
Thanks again for everyone's insight on this matter.