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If you can get a CSP you can get an Amex revolver.
Charge card if desperate in your situation to set MSD but frankly you've got 10 months before needing to worry about that so apply for the card you want.
Charge card undoubtedly helped me but my FICO's were ~630ish at the time, 640 Beacon 5 by the time I applied for the revolver. Good chance my FICO 8 scores were higher but I didn't have access to any of them at that point.
Recon may involve an additional hp. In the case where the CR was erroneous they may pull to verify the update or if they pull from a different bureau in an attempt to approve a borderline application.
Bottomline is a reconsideration may involve an inquiry.
To answer OP's question, yes, a revolver (when used correctly) will help your internal score with AMEX, thus helping you get revolvers down the road. call the EO. they're super nice and very helpful when reconning.
I'd only go for the charge card if there's a really good sign up offer like on the PRG.
Otherwise, just go for the revolver you want and build some history off that and you'll find it so much easier to get other Amex cards if you give it time. Just two months after I got my Delta gold, I was able to get the Everyday card and a few months after that the BCE and last month the PRG and Hilton HHonors cards.