Opportunity is missed by most people because it is dressed in overalls and looks like work. - Thomas Edison
I won't use just a low bleed for any blended or 100% poly tee, we only use the no bleed poly whites. I've found that even the low bleed whites will bleed due to high dryer heat so I use Rutland Poly white, Xenon Horse white or Triangle Excel for any shirts that may bleed. I screwed up a 700 piece 50/50 red shirt job using a low bleed white as an underbase and top white and the ink failed so I don't even try any more.
This is one reason why I have not left Quick White. We have had none of the issues above. No bleeding. We use poly white on 100% poly garments everything else gets Quick. 50/50's, 60 poly 40 cotton, and tri-blends. Never an issue. Those cheap whites just are not worth it.
OP states that the garment is not a blend and is garment dyed which can also bleed like a pig especially if they use sulfur dyes. What is the label on the garment? Chouinard perhaps?