Dang, I gotta remember to check the box for being notified of replies by email
Maybe you got it figured out by now but if not, I think you're on the right track.
Three ways to go about this (making a short slab of end-grain wood to form a new lip):
1.) Cut strips across the end-grain from your chosen lumber, and glue them together into a square big enough for your box's diameter as you mention in your last post.
2.) Glue up a segmented ring, 6 or 8 segments, from your lumber. I would rip just one strip on the tablesaw and cut that into 6 or 8 short pieces to form the ring. Set the saw at a 30 deg. angle for 6 segments, and 22 1/2 deg. for 8. To check the angle, run a piece of scrapwood through the process first. Cut it into small pieces, and glue together with CA glue--if you end up with a gap between the first and last segment, increase the blade's angle a tiny hair. If the space for the last segment is too small, decrease the angle (again, just a hair). A trial-and-error process that may take a while! Once you get the angle right, use Titebond to glue the final ring.
3.) Cut the lip from a solid slab of wood. This is what I do, I have preturned cylinders of various woods ready to cut a short section from. These are all from green wood, so they need to be dried before gluing. I use the microwave. I cut out the center first to make a thick ring, this minimizes the chance of cracking.
Befor gluing the lip or tenon onto the box, the mating surfaces must be flat. I do this mostly on the lathe, with a light shear-scraping cut. Use a straightedge across the end, with a light in the background to check for flatness. If I'm just a hair off, I'll take it off the lathe and rub across a sheet of sandpaper on a flat surface.
I glue the assembly on the lathe, using the tailstock as a clamp. After the glue sets, I may rough-turn the added piece a bit, but will not do any finish-turning until the glue has cured--at least a couple days.