TheBarqsMan (see also here) has given me the bad news regarding my earlier quest for a local source of caffeine-free Barq’s. He says

the only distributor of Caffeine-Free Barqs [is] Swire Coca-Cola in Salt Lake City, Utah. They distribute to all of Utah, Northern AZ and a few places in Idaho and Washington.

And checking their territory map leads me to the sad conclusion that they don’t bring the stuff any closer than the Tri-Cities.
Thanks, TheBarqsMan!