For a short amount of time, I don't think there'd be any problem with running different tires as long as they are measured to be the same circumference. DO NOT just go with 2xx/xx/1x on the tires and assume theyre the same. My RE-01R's lined up next to my old Hankook RS2's are each about a half inch wider.
Of course it's not a good idea running different tires over a long period of time. One will wear out quicker than the other, and since you want the RE070's are gonna be on the front, they'll most likely wear significantly faster than the fronts. As for the diffs, assuming circumference is relatively close (aka within spec), I don't think the different grip levels and such are gonna put that much of an extra strain on them. If that were true, anytime any of us understeers or oversteers, our diffs would blow up.
CN: Short term you'll be fine, but unless this is just a temporary idea, it's dumb.