I don't know about every bank, but Bluebird bill pay for small (less than 1000) amounts posts to Chase late the next day for me, and the available credit recovers as soon as it posts to the CC.
Have you tried calling the number on the card? If you explain the situation they may have another way to allow a payment over the phone after you reach a CSR. Citi has even let me pay with a debit card to free up credit to for purchases, and that posted and gave me the available credit before I hung up.