I agree that the surest bet is a stallion proven for running barrels. However some barrel horse breeders, (and even cutting horse folks believe it or not) have in the past and continue to infuse some Thoroughbred blood from time to time. Usually they use a proven barrel horse (or cutting) stallion over a Thoroughbred mare. But not just any TB mare-- the best risk is a sprinter type, rather than a long distance runner-- for example I have a TB mare who ran most of her races from 1/2 mile to 3/4 mile-- and was winning with fast 1/4 mile split times. She is not that tall (15.2) and has a wide body, and more muscle mass than a distance running TB would have. Thats the type of TB blood that can add some power for fast takeoffs and turning. However it is a risk, and it also is not something you can do and expect everyone to come running to buy the foal-- a foal bred like that would not be immediately marketable but would have to grow up and show its stuff before it was as marketable as a foal from proven lines on both sides for the chosen discipline.
And yes, if your horse is a "regular" (not appendix) registered QH, registered Paint, or regular-registered Appaloosa, you can cross to registered TB and the foal will be registerable in the dam's registry.