Briggs sell a grab pack of connectors and even more important transfer plugs to plug in different styles together.
So any B & S REPAIR center should be able to fix you up.
Take your original engine number and new engine number with you or photos of both connectors.
There are a couple of B & S Master technicians on this list who shoukd be able to either tell you the B & S Number or sell you the right bit through the mail.