I don't think it would be too hard. After the round players throw their cards in by suit face up to 4 piles, every suit gets topped up to 12 cards. Have one person take all four suits under the table and then hands them in random order to the dealer. The dealer randomizes them themselves so nobody can know which suit is which, then removes the required number of cards from the random suits and discards them, and shuffles the deck.

Obviously it isn't super quick, but once people know whats going on it doesn't take that long either.