No, the point of DMCA is the provider is agnostic to who owns the content. Its the making a decision on who owns the copyright that opens them up to litigation, because they would lose safe harbor protection. Incidentally, NOT putting the content back up would be a violation of the DMCA, which also opens you up to litigation.
"The person who filed the complaint is a mailinator address, or in Russia/Iran/Cuba or works for a corporation with more lawyers than you."
So they're either a 1) liar, in which case your content is back online and you're back where you started or 2) a big corp who can threaten to get their way? Welcome to the fundamental problem with the DMCA: it can be abused.