As far as my researches have shown, side clips were never the subject of a patent. I don't think they were even illustrated in a patent drawing but I may be wrong.
Purdey, Woodward, Boswell, in fact anybody who had powerful guns for heavy loads used them and I don't think that Purdey stood out as a particular early user. I have seen very few Purdey hammerguns with sideclips but plenty of lesser names.
I suspect that they became a required feature once someone won a big bet/prize with them fitted and everybody followed suite, including the most illustrious. They simply supplied what the customer demanded, especially in the live pigeon matches.
Yes, Purdey patented a couple of very important mechanisms but I doubt they were interested in a feature which frankly achieves very little for a shotgun (arguably more important in a double rifle). They just followed the fashion.
IMHO