I say call in and speak with the CLI dept specifically. 800-927-0632. You are asking for a sizable increase. For example going from 2k to 6k is wayyy easier then going from 12k to 36k although both are the 3x CLI. The "3x rule" is a nice guide but doesn't always apply.

My experience with AMEX is they usually try to accommodate you if you keep your payment record clean. Good Luck.