Solution:
I sat down, thought things over, and decided I needed to mimic that "pulsing" effect of the cleaner to break loose whatever was in that jet.
I took 100% simple green, and pulled it into a standard 10ml medical syringe. The tubing I used is a 3/32 ID gas line...the kind you use for a chainsaw or mower. The tubing fits almost perfectly over the post for this, no need for a twist-tie with this method.
Filled the syringe as shown here...you want that air space.

Then, placed the tube over the post. I noticed that i had to push the tubing down into the carb, putting a link in the line. This pulled the tubing away from the face of the post just enough to allow the fluid in the line to get between the tubing and the opening in the jet:

From here, simply pushing on the syringe does nothing. The pressure builds up and blows past the line at either end. What you need is a sharp, sudden increase in pressure. Hold the base of the syringe I'm one hand, and pull back the 'plunger' with the other. You should feel the backpressure pulling back A LOT.

Release, and it should snap down. After several repetitions, the simple green became foamy, and I noticed bubbles streaming from the post. Next pull of the syringe, and there was no backpressure.
Jet cleared.
