"I will be using the Enter Credit Card Charges/Refund for posting the Amex charges to the appropriate expense category in Co A. The rest of the entries will be JE. Will this be ok?"
Company A enters the Charge. They also enter Credit Card Credit posting to Other Asset for the Amount Company B paid directly to AMEX. These get their separate Dates. Now the Other Asset account = 0 = Repaid in full.
Company B can simply Pay AMEX and list the expense right on the Check to pay Amex.
Company A pays their AMEX balance by Checking, listing the AMEX account as the "expense." You typically do this after using the AMEX statement to Reconcile. That is not a BIll; it is your Account Statement.
There are no JE involved in any of this.
