You could just leave it down. If some urine gets on the seat, that's what toilet paper is for.
'course, as men we tend to wake up with certain "protrusions" that tend to intefere with maintaining a direct stream - thus necessitating the use of tremendous amounts of TP (or a mop). I like to call my solution to this problem "the Superman" - basically, you lay across the seat and fire directly into the porcelain target...thus preventing the necessity of TP.