Patience is key. You may first want try to put the barrel into the refrigerator (within a ziploc bag) to make the wood shrink a bit. Then warm up the ring in the round of thumb and index finger so that it expands a wee bit, this helps getting it loose.
If the ring in pushed all the way in, you need knife with a sharp blade. Push the blade sideways (don't use the tip) between tenon/barrel and ring, just so that the ring is lifted a bit. DO NOT WIGGLE. Now do the same at the opposite side. Gradually take knives with thicker blades till the ring comes free. Make sure you're not using excessive force, and don't push too hard, else you cut the ring tenon.
If the ring is only part way in, use the back side of a dinner knife (blunt), or two of them, or the handle of a spoon...whatever is blunt and will apply an even force onto the wood, to avoid nicks in the wood.
If you find that while you push the ring on one side up, it's getting pushed back at the other side, use toothpicks or sharpened matches as wedges on the "idle" side.