Does anyone know whether Beesley made any of his own guns with his patented self-opening mechanism, or were they made only by Purdey?

Yes, Beesley made self openers of his patent under his own name

Atkins also made self openers in the same style as Purdey after the patent protection ran out

Crescent Firearms mfg'd the Triumph Hammerless.
https://patents.google.com/patent/US320040

Many makers have made guns on the Beesley Action. Purdey by far the most, but also a few hundred by Atkin, and a few by Dickson, AyA, and many others.
Apart from Purdey and Atkin, no other maker I'm aware of has made more than a few (AyA are believed to have made about 40).

Beesley had two patents for self openers. 1 sold to Purdeys the other sold to AA Thorn. I’ve seen examples of the latter with Beesleys name marked on them, very rare.

Beesley made several Beesley 1880/31 Patent guns to match Purdey guns on bespoke order only. Some of the reasons: According to Dallas or Beaumont, I forget exactly the reference, James the Younger had closed his books for the year. He wasn't taking any more orders. So they went to Beesley at 2 St. James and his crew on Pickering Place made them up. Some people wanted a gun to match their Purdey so they went to Fred. Maybe some liked dealing with the Beesley organization. It was not for price, because FB often charged more than Purdey did!

Scotts had the license from Purdey for the 1880/31 patent for Birmingham guns from Purdey. Francotte had the continental license from Purdey. I saw one of these at Stephan Hutton's that I think came out of a Julia auction. I have pictures of it and it will be in the book. Francotte also feature the action as their top design in many of their pre-war catalogs. There is more to the story.

You have the only one I've ever seen.

There have to be a few more circulating. Mine is ser# 762.

http://www.doublegunshop.com/forums/ubbt...8214#Post488214

Bass had a cased Beesley SLE, SO at their Springfield, MO shop discounted due a cracked stock. It did not last long.