I think you will get more or less brown tint if you breed a RIR to a white egg breed. but not quite white.
The general rule is that the eggs will be an intermediate color to what is expected in the two parent breeds.
In fact, comercial white leghorns have a gene to "clean up" the tint. (But white leghorns are useless for color sexlinks)
I'd get some California Grey or Production Black pullets - redily available, barred, white egg layers.
You'll get sex links from your RIR rooster, which,I think, will lay much lighter, if not quite white, eggs.
For sex links to lay a green egg ... if you could find some silver americauna
For darker brown if you could find cuckoo Marans that actually lay darker than RIR you could try that
or any extreme dark egg breed that has the S or B gene