Not the Met area myself , but , There should be no charge for adding a shotgun to a certificate . The process is this .
The seller of the gun writes the gun make , type ( ie o/u SxS etc ) , calibre and serial number on your certificate ( you don’t do this ) , together with their Name , Address , and authority to sell , which will be either their SGC if you’ve bought it privately or their RFD number if they’re a dealer .
You then ( within 7 days normally ) either complete an acquisition form by hand and post it to your licensing authority , or complete an online acquisition form for which you will receive a confirmation e-mail .
The handwritten entry by the seller is all you will have on your certificate until your next renewal .
I think the £20 fee you refer to is a firearms variation fee , nothing to do with shotguns .,