When you pay electronically, that is a Paperless, virtual check. If you don't want to use the Check function, then I recommend this method:
Enter Credit Card Charge, but change the radio button to a Credit. Select the Bank account on the Expenses tab. This shows that the Bank account was used to pay this balance against the CC account.
"Under the account listing, I can see both the store credit and payments listed under "Payment","
Do you mean, when you view the CC account as a Register view?
And as long as the JE is posted to the CC account, and not to the expenses, then it will show in the CC account. Confirm how you listed the entry on that JE. No, it is not the best transaction for this, but if used correctly, it should show and reconcile just fine.
Did you enter the correct Ending Balance from your statement?