Trade hats? Never. The hat is totally Urahara's trademark. I swear, he probably had "the key" that Aizen is looking for hidden underneath it or something. Not to mention, Ulquiorra doesn't exactly have a "hat", more of a shelled.. thing.
Sorry, but no trades. Without it he's just not the same man. Ulquiorra couldn't handle it anyway. Only Urahara could possibly manage both Soifon and Yoruichi at the same time for he is sex incarnate.