Just because you have high limits on other cards does not necessarily mean Amex will approve you for a similarly high limit... especially if your available credit far exceeds your income. Also note your Amex Platinum card still has a limit... they just don't tell you what it is. You only find out when you try to make a purchase and you exceed what they're willing to let you charge, and your card is declined.

Nothing you can really do to increase it for 60 days. On day 61, Amex generally allows you to request an increase up to 3x your original limit - in your case $15,000 - though approval is not automatic or guaranteed. You can then apply for another CLI every 6 months.

Amex generally requires a 4506-T if you request an increase that would put you over $24,500 for any single card or over $34,500 across all your credit lines.

As others have noted, you can pay mid-cycle and spend far more than your limit every month until you get your CLIs.