How would that work? If I were to cross the Roo with let's say an RIR would that leave a possibility for a chick with barring over the red?
		
		
	 
How the barring gene works:
Barring is on the Z sex chromosome. Males have ZZ, females have ZW (yes, backwards of how mammals do it.)
The barred hen gives a Z chromosome to her sons (with barring), and a W chromosome to her daughters (cannot carry barring.)
The son has two Z chromosomes: one from his mother (with barring) and one from his father (probably does not have barring.)
When that son produces chicks, he gives one Z chromosome to each chick. About half the chicks get the one with barring, and the other half get the one with no barring. The gender of the chicks will be determined by whether they get Z or W from their mother, so when the father has barring it can appear in either gender of chick.
So yes, that son could produce chicks with white barring.
But you will not get barring on RED chicks from that rooster, because he is pure for black (gets the dominant E gene from both his father the Ideal 236 and his mother the Barred Rock.) Because he is pure for E, he must pass it to every chick he sires, so they will all be black. Half of the chicks will also get Dominant White from him, and half get barring, so the total distribution of chicks should be like this:
25% black with white barring
25% black with no white barring
25% white with white barring
25% white with no white barring
(The whites will look alike, but whether they have barring will affect what chicks they can produce in the next generation.)
If you use a cockerel who has a Rhode Island Red mother, he will not have barring (unless the Ideal 236 rooster carries it-- and we have no way to tell if that white rooster also has white barring.)
But the son of a Rhode Island Red mother will not be pure for E (which makes chickens all black.) He will be E / E^Wh.  The E (=all black) comes from his Ideal 236 father. The E^Wh is called Wheaten, comes from the Rhode Island Red, and allows a chicken to show some red as well as black.
So a rooster who is E / E^Wh will pass E to half his chicks (black, or turns white with Dominant White). He will pass E^Wh to the other half of his chicks, causing some to be red-and-black and others to be red-and-white (because Dominant White turns black into white, but leaves the red alone.)
The black & red may not quite match a Rhode Island Red (red with black tail), because there are some other genes involved in how the colors are arranged. But the basic colors should be present, rather than just producing black chicks or white chicks.
If a rooster has Ideal 236 for his father, RIR for his mother, and is mated with a red or gold hen, his chicks should be:
25% black (E, no Dominant White)
25% white (E with Dominant White)
25% red-and-black (E^Wh, no Dominant White)
25% red-and-white (E^Wh with Dominant White)
If you use a rooster with a Buff Orpington mother, you will get something similar to what the Rhode Island Red gives (but likely with gold rather than red.)