You should rather ask: "Darf ich Sie bitten?"
"Ich darf Sie bitten" seems odd, and, if at all, outdated. In my view, a sentence like that would imply authority over the person addressed. It would be a polite form of what would really have to be understood as a command.
Also the "Darf ich Sie bitten?" sounds a bit outdated to me, but might be used in an ironic, cyncial or aggressive undertone, e.g.; "Darf ich Sie bitten, sich an die Fakten zu halten/ mir etwas genauer zuzuhören? o.Ä.
Without any verb this question can also be used at formal dances, when a man wants to invite a lady to dance.
Please also note that "Ich bitte Sie" can have a wide range of established meanings depending on context. In a lively argument it may actually not be meant as a kind of request, but rather as a call for reason when one disagrees strongly. E.g.: "There's no way you can argue that Donald Trump is a trustworthy human being, ich bitte Sie!"