I don't think being out of UG for one year is enough to justify not having LOR's from professor.
I wouldn't worry about it, or better yet, volunteer to help your supervisor edit it.
I think it is a good one to save for waitlists. My employer of six years wrote a great one that was extremely unique and compelling, so I sent it in for waitlists.
He also owns the restaurant where I have worked for six years, and where the law school that has waitlisted me frequently brings visiting professors. Wanted to get his name in there somewhere .