Like half of the deathflags poibting at his head would be immediately lifted if he spoke with his brother about Wilhelm, but let's go with the japanese approach of expecting the other party to read minds instead of saying it out loud.
Oscar isn't his ally, though. More like an enemy. As long as Rudolph belongs to the Fredrick faction, he's actually one of Oscar's main adversaries. Fredrick was lacking all military backing, unlike Oscar, but Rudolph brought that Fredrick. So, now Fredrick has both political and military backing, whereas Oscar mainly has military. Oscar either needs to get Rudolp to switch sides, which is pretty difficult, or destroy Rudolph's alliance with the Fahlkrantz family, which would make Rudolph worthless for Fredrick. I have no doubt Wilhelm has privately indicated he would help Oscar, especially in this matter. Wilhelm needs the princes to utterly destroy each other, after all.
Things being as they are, Rudolph is better off not revealing his intel to Oscar, as Oscar wouldn't help him anyway. Even if Oscar nominally kicked Wilhelm out of the student, he would still maintain his ties with Wilhem in the background. It's not like Rudolph could simple tell Oscar that Wihlem is planning to destroy the whole empire to found his own kingdom. Pretty hard to prove such a claim. On the other hand, that could also be considered normal ambition for important nobles, simply never stated aloud.