Ok...where would I find this?

Mini Bike & Go-Kart Parts

Is there an adapter that would allow a smaller sized barrel to fit into a 3/8" barrel hole in the handle?
Yes, they are called slip barrels, you may have to modify the barrel that comes on the cable to fit it though as this type is mainly used for adapting ball end cables to 3/8" barrel ends. The smaller end cables are also smaller wire so keep that in mind too.

I did email Flanders. Waiting for a response
PIck up a phone and CALL them. I have found that many companies are very slow to replying to email inquiries, and some never reply at all. When I have CALLED asked them about this they generally reply that emails are low priority and they get around to them "when they can."- just saying...
If you can find the smaller cable in the length you need locally, cut off a 3/8” section of a 3/8” grade 5 bolt, drill a small hole in it for the cable, then drill it out half way to 1/4” for the barrel end. I would see if I could find a ball end cable first or grind the barrel so it’s round if you can’t find the ball end but I guess if you could do that you wouldn’t be here asking for help.