If it is correct that James PERKINS married Ann* RAMAGE in 1882 (not married Ann* PERKINS surly) then the only way I can make sense of the death certificate is thus.

The informant John SHORT may have been son in law or grandson (in law). It is quite likely that SHORT was not filling in the form himself but answering questions.
So when SHORT was saying that the father of the deceased was James H PERKINS and the mother was Ann RAMAGE he was confused as to the question. He answered as if the question has been what were the parents of a child of the couple.

So the "info" that the father of the deceased James PERKINS was called James H PERKINS is wrong - It is just saying that the deceased was called James H PERKINS born in GLS.
Similarly the "info" that the mother of the deceased James PERKINS was called ANN RAMAGE is wrong and it is saying that the wife of the deceased was called Ann RAMAGE born in Fifeshire? Scotland.

So we know nothing about the parents of the James PERKINS born around 1860 and cannot reject any whose father was not James and mother not Ann.

As regards the middle name starting with "H" I note that one of the newspaper mentions of the death have him as James S PERKINS so I'd not put much weight on what middle name he had if any. It could even be that when born his middle name was James. Given names are commonly swapped.

Though people can get the years wrong they normally remember the day and month of their birthday and the records seem consistent that he was born on 16th April and so his birth should have been registered in the Jun qtr.

Looking around 1860 in Freebmd the only possible seems to be the previously mentioned James George PERKINS whose birth was registered in Bristol district Jun qtr 1859 vol 6a page 49. If you get a copy of that birth certificate and it has the birth date as 16th April then it must be your man.
If you do not want to spend £9.50 then apply for it but do not give the vol and page number. Tell them the name, quarter and year and say you only want the certificate if the birth day is 16th April. If it is not then they will not send you the certificate and not charge you.
I think that's the way it works. Personally I've generally just gone ahead and bought the certificate. Can mean a collection of not quite right certificates but means that I have no lingering doubts.

Bryan

ps
Another possible birth registration is
Births Jun qtr 1858
Perkins James Evesham 6c 324
(the Evesham district covers quite a lot of GLS