A lot of churches, especially older ones, don't allow animals in them.
While I wouldn't be surprised to see some modern touches, I don't see a 'dog ring bearer' or such a modern concept at a senior royal wedding.
I am not sure if St George's will, but I wouldn't be surprised if they had an unique piece commissioned for them. William and Kate were not the only ones. A fanfare was commissioned for the wedding of Charles and Camilla, by a Welsh composer. It was played when the couple left their blessing. The composer composed music for Charles twice before, for his investiture as prince of wales and on his 16th birthday.
Welsh Composer Writes One-Minute Fanfare for Royal Wedding | Playbill
It can be heard around the 5 minute mark here.