Well arguably the grain is the most important part in dog food for the roaches, seeing as there are studies citing the uric acid from high protein cat foods can cause health problems in certain roaches, (not that the average hobbyist typically comes across such problems though), and I've been using unmedicated chick feed as a replacement for dog food for a while now with no issues, (all it is is just grains, no meat).
Rat chow should actually be perfect for them, and honestly grain free dog food will likely attract just as many mites. The problem isn't the food, it's likely just the amount you are feeding them, if there are regular leftovers, I'd suggest feeding them smaller amounts, and place food in bowls if you can, as that can help prevent grain mite explosions. You can also try keeping the feeding areas drier than the rest of the enclosure, and adding springtails to the setup to compete with the mites.