I don't think know if there's an answer for your question on the Internet. Just did a bit of googling and find under "special marriage license" the following as applies in Jane Austen's time:

"A special license was the most expensive type of license, costing between 4 and 5 pounds, and were issued at the discretion of the Archbishop of Canterbury. A special license permitted a betrothed couple to marry at a time and place of their convenience, though they were encouraged to marry in churches. The only people who could obtain special licenses were peers and peeresses, their children, baronets, knights, members of Parliament, Privy Councillors and Westminster Court Judges. Fitzwilliam Darcy would not be eligible for one, though his cousin Colonel Fitzwilliam would be."

Cited are three excellent sources, Pat Rogers, ed., Cambridge Edition of P&P and David M. Shapard, ed., The Annotated Pride and Prejudice, and of course, Jane Austen. Jane Austen for Dummies is also cited.

A history of special licenses can be found here:

https://familysearch.org/learn/wiki/en/Marriage_Allegations,_Bonds_and_Licences_in_England_and_Wales

It's clear from this reference that the availability of special licenses differed considerably over time and there is no specific reference to the Regency Period. Google does provide hundreds, if not thousands, of examples of special licenses
historically and at the present time. The best search terms seem to be "special marriage license."