I am aware that the open register has an opt out option, but does the full register also have an opt out option?
The gov website is silent on this issue.
You can't "opt-out" of the full register. You can decline to complete the registration process. But the local Electoral Registration Office could fine you £80 if you don't register when asked.
You may be allowed to register anonymously.
The full register is an authoritative list of those registered and allowed to vote in elections. It is the list that is used to produce the polling cards that are sent out calling you to vote and by the officials at the polling station when they record that you have voted (obviously not how you cast your vote). If you "opt out" of this list then you are not registered and cannot vote. You can "opt out" of this list quite easily; just don't submit your voter registration in the first place.