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Amex will have no issue internally with your balance; it shows that you're making use of the credit line and will suggest to them that you need more. It is exceptionally rare for American Express cards to report before the 2nd statement closes. Since we're 99% sure that it won't report to the credit bureaus, I wouldn't be too worried about it. If your bank does not allow you to push payments, you could always mail a check. How long until the statement closes? Once you get past the first statement closing, you should be able to pay as many times as you'd like.
Don't sweat it. Let that balance post and then cut the payment is what I'd do.
End of the day I'm still of the opinion that *using* one's cards is the best thing you can do for future approvals... and a high balance shows literally everyone that you not only used it, you used it responsibly (assuming some future balance is much less and no lates / default).
Might not matter, we've been living in the credit sunshine for nearly 7 years now and trended data will come eventually probably everywhere... but there's cracks in the credit markets which seem to be showing now and I think it's best to accrue any small advantage one can. Bank's like profitable customers, show them you at least might be and that's exactly what a non-trivial high-balance can do.
If you got it in the mail tomorrow, it's certainly possible that it could be received and processed by next Friday, but that is cutting it close. Let me put it this way, if Amex had real concerns with you, they would never have allowed you to cycle through your entire credit line twice in the first month, allowing you to get up to 100% utilization twice already.
The one payment thing is just a fraud mitigation thing for them. Their reasoning is that since you're a new customer, they need to allow time to make sure that the account you're paying from is legit and that they're not going to be hit with reversals. Someone please correct me if I am wrong, but I think the multiple payment thing happens with any additional bank accounts added for pull payments.
Seriously, what sort of bank are you using that wouldn't allow you to push a payment via ACH to Amex?