I'm sure the RNC will compile a report that going after illegal voting will hurt the RNC's chances with Hispanics and that this is the future of the electorate in the United States...(which seems to be the conclusion of every RNC report recently.)
Well maybe they will, but that doesn't deny the fact that they have the information. Besides, it's a crime, that would make them complicit. If the President wants it, he will probably get it.
He could simply threaten to ask his supporters to leave the party.