"I've lived in the DFW area for 17 years and have never heard KAIJ (a shortwave broadcaster) located here."
Where is "here"? They claim to be in Dallas TX while the mailing address is Murfreesboro TN and the transmitter site could be just about anywhere. Unfortunately I don't have the proper FCC URL for a lookup or I'd nail it down for you in lat and lon.
"I assume their radiation pattern does not favor local reception."
That depends on how far you are from the transmitter, what frequency you're listening on, and an infinite number of constant variables to borrow a phrase. Without a polar plot of their coverage area furnished by the engineering firm that did the original site survey it's impossible to say. All in all I'd say you're in the skip zone since if you were in the path of the direct wave you'd hear it to one degree or another no matter what the pattern is, no antenna system gives perfect nulls. In other words if you're anywhere near that 100KW transmitter you'd hear it.
BTW, I get a chuckle out of the way those Bible thumpers and broker jokers dance around the FCC's famous shortwave loophole. The rule says the target audience must be outside of the country but the programming says that just ain't so. Brokered air time obviously targets a US audience so the transmitter site, frequencies, and antenna patterns are chosen to blanket as much of North America as possible, the foreign audience is but an afterthought. You may have noticed these guys aren't megawatt flamethrowers like the VOA, 50KW being minimum legal power they barely qualify as fleas on the back of short wave international broadcasting.