How would a native British speaker say "Betteredge"?

Most British speakers would neither voice the '-tt-' nor devoice the final '-dge'. They might however syncopate the middle syllable. So

/'bɛtəɾɛdʒ/ or /'bɛtɾɛdʒ/ (better-edge or betredge)

They might raise the final vowal a little

/'bɛtəɾɪdʒ/ or /'bɛtɾɪdʒ/ (better-idge or betridge)

Proper names in English come from such a wide array of sources, and go through generations with relatively random dialects. The result is that you pretty much cannot make any assumptions about how they are pronounced. You have to listen to the person say their own name to know for sure.

Being the cradle of the language, England is the worst about this. How the same name is pronounced can change radically multiple times just within a few city blocks in London.

For this reason I really don't worry about proper pronunciation in situations (like reading works of fiction) where there's no way for me to find out the right answer. Just come up with your own and move on.