Shooting legal advice
A shotgun certificate variation will not be needed. A variation on your licence is only needed if you want to fit a silencer (usually called a sound moderator) on a rifle.
No certificate is needed for a silencer on a shotgun or air rifle but it’s difficult to fit a sound moderator to anything other than a single barrel shotgun.
Probably the best thing to do is to buy a gun with the moderator already fitted. Such guns are typically semi-autos and are relatively inexpensive.
They tend to be bulky and unwieldy but do reduce the noise of the shot quite significantly.