Well, it's an Adventure Cycling Route, have you looked there?

Would you really be going from Flagstaff to the Canyon, since the South Rim is west of Flagstaff? It seems like if you're going west-to-east, it would make more sense to use the Adventure Cycling route on the way to the Canyon, and then go through Cameron->Flagstaff on your way out.

I biked in and out of the North Rim in spring, so the Cameron->Flagstaff section would be common between us. From either rim, Cameron is the low point, so you descend a big hill to get there and then climb another big hill back into Flagstaff. The Cameron area is mostly blasted desert, which gets nicer as you return to elevation near Flagstaff along US-89. The western approach on AZ 64 is probably a bit nicer, as it stays at higher elevation the whole way to the rim. Either way you can expect a whole lot of nothing, but that shouldn't be out of the ordinary for you by that point.

Neil