I'm not really sure what you've got set up. If the company is reimbursing you for all the CC charges, then it is just a Check written to you (or a Transfer) and that check get shown as the Owner Draw you're looking for. Then you show that as a deposit in your own Personal Acct and write a Personal check to pay your Personal CC.
If you want to run all the charges through the Business and pay with the Business Checking, then set up a CC Account in QBs. For all the Personal charges in the CC register you would associate them with Owner Draw to show they are not a Business Expense. End of the month you just write a company check to the CC company.
Personally, I'd get a Business Credit Card and be done with it. I saw that you said you didn't want one, but I can't think of a single reason why not. It removes all Personal charges from the Company records (something you should strongly consider) and keeps your Business transactions out of your Personal finances. I'm sure your Tax Accountant would love you for this.
