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With your profile, I can't see why you couldn't just do both.























I'm not sure if it really matters which order you do it in. DW asked for a 3X CLI on her ED card in the fall to $36k and got a counter-offer of $32k, but then when she apped for the Delta Gold SkyMiles card a few weeks later, her starting line was $23,700. I thought it was odd that they didn't extend her the additional $4k she asked for, but had no problem giving her another $23k+ on a new card. I bring it up because even if you do the CLI and they don't give you the full amount, that doesn't necessarily mean that you're tapped out with Amex.
Did you burn Amex in a BK previously or still have one showing on your reports? Just asking because the $500 and credit steps seems odd for your score (unless it's changed drastically since applying).
Given the DP I've seen here, do the CLI first. It might mean a higher SL on the new card. When I apped the Gold Delta Skymiles, i still had a low limit on my BCE. And in turn didn't get a very high SL. So when it came time for a new card with Citi, i did my CLI first. Then apped for the new AADvantage card. Got higher SL than I thought I would.
Off topic here to the actual question posed by the OP, but something worth mentioning. You mentioned that your newest account was opened on May of 2017. This means that on May 1 of this year your AoYA reaches 1 year. That's a significant benchmark and could easily result in a 10-15 point score gain. Outside of the score gain, it's simply a great look when applying to have an AoYA of over 1 year.
For you, that's 1 week away. I'd honestly wait that week before apping. Will it make a difference in your approval terms? Maybe yes, maybe no, but it can't hurt and IMO it's worth the week wait. You can knock out the CLI request in the meantime and know that a week from now your profile will be even stronger, giving you greater confidence in your app.
Seems to me that you should get a card that has good gas rewards!