Separate account, separate fee.

If you are a principal of a closely held business I cannot see a compelling reason to have both Personal and Business Platinum charge cards unless you need additional Platinum cards for other household members. I would have the company pay the annual fee for a Business Platinum charge card, which you can use for benefits, and use a low or no fee personal American Express card for personal spending. If you do put some personal spending on the business card you can link a personal bank account to it to pay for those charges, or you can reimburse the company if your accountant prefers.

If you want to maximize Membership Rewards the personal card could be Zync or Premier Rewards Gold linked to the same Membership Rewards account. If you prefer to diversify your rewards the personal spending could be put on an affinity credit card such as Starwoood or Hilton.