I am answering original posting (OP) question.
"I know one solution is to suggest when the owner knows she wants to pay the AmEx card with Company B's checking account she writes a check to Company A for the payment amount, deposits the check in Company A and makes the payment from Company A's checking account."
I agree. This is indeed a perfect solution that will show an arm's length transactions between two companies.
"Since this involves extra work for her she may not want to do this, so as an alternative ... an easy way to make the entries in both company files."
If I were to do it I would just write a simple journal entry. Other ways to do it in QuickBooks as well.
Company A: Debit AE account; Credit current liability account.
Company B: Debit current assets account; Credit bank account.
(This example entry is when company B making payment (full or partial) directly to AmEx on Company A credit card. There is no actual cash transfer between Company A & B at this time. On company A's books, it will clear credit card balance but will show due to Company B as Other Current Liabilities type account. On Company B's books, it will show Due from Company A as Other Current Assets type account).
At some point, you will need to clear these intercompany balance sheet accounts.
There are other alternatives (already addressed here) but as you said: "she may not want to do it."
