I hate to discourage you, and it does sound like a really cool idea, but for etiquette reasons I would shy away from using the badge as a ring. For starters, it is only supposed to be worn at very formal occasions - generally, people wear their rings all the time. Second, there are very clear rules in The Paedagogus about when and where it can be displayed, and I doubt a ring would qualify.
If you're set on getting a Lambda Chi Alpha ring, and keeping within the fraternities' traditions, I would suggest finding a lettered ring or jewel on eBay, and having that set into one, or getting one engraved.
Hope this helps, & ZAX
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