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I've done that with Citi and they agreed to move my date. They only let me pick either first half or 2nd half of the month though and not a specific date.
Getting all the due days in a cluster is best. However, I am making an exception to that rule, separaing my Walmart and Discover cards so I get a different, brand new FICO every two weeks. But, I think your idea should be the general rule.