Just for giggles, I threw this splice together with the ring in place using some Ice Tail I have laying around.
You do the first locked Brummel 'exactly' as if without the ring. However, insert the ring for the second pass, then add a third Brummel with the short rope end BEFORE burying the end into the core.
Note: For reference, I intentionally left a larger loop in the last Brummel end as it passes into the core for final bury.
The only caveat: This works because the eye will be girth-hitched around the chainsaw. If not, the rope at the eye would be pulled through where the ring is when weighted.