Being a pedant hasn't it changed its name? It was originally called the Bear Inn and is now called The Bear.
I can categorically state that The Sir Charles Napier has always been called The Sir Charles Napier, so I think The Bear's website is telling a few porkies don't you?
I think you'll find that is not true, the pub I grew up in was called The Sir Charles Napier when it was built and is still called it now.
I think you'll find plenty of pubs in Brighton and Hove with their original names.
The left one is where I grew up and now have the occasional drink with Eric Potts.
The middle is five seconds from my front door.
The right is where I go to drink Harveys and Timmermans Frambroise!