Correct. You can take it as a legal Detachment, even with the beta faction rules that don't permit using CHAOS as the detachment keyword. Note that that rule is beta, and therefore might not be in use in some tournaments anyway. As you say, you won't benefit from certain Death Guard faction abilities though.
You are also correct about the relic.
With all of this though, it's worth checking the specific guidelines for the tournament you are entering, as they may differ from the official rules.
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