I spent a long time fighting with my lathe today... But I finally got an alright solution. I visually turn it down to just over half width with a dremel stone in the lathe chuck, with the ring clamped to a bit. Then, I used a file to gently and gradually bring it down to as close to halfway as my measurement tools/eyeballs can tell me. I found that the best way to do this is to measure the overall thickness of the lap instead of each half individually.
I don't know if it's an issue, but the corner of the cut on each edge of the ring kind of sinks down a little bit from the edge due to my imperfect filing abilities, but after I looked at the stock ring, noticed there was a bevel, and added a similar bevel to mine, it seemed to fit quite well. Whaddaya guys think, I don't know how much better I could get at this with the limited machining tools I have. I don't know how important perfection is here, but something tells me some imperfect lap is better than a gap. Edit: The filing looks a little rough, but of course I plan to use a whetstone to make it nice and smooth.