It should be up to the doc, not the patient to decide what's best! The decision should be based on fact, not simple unstudied preference!! Good grief, didn't you have any testing done? Doesn't your doc make the final decision during surgery when he sees what's available to use?
My wrap is a Toupet, based on my very poor swallow - only about 40%, and not being able to dissect down enough esophagus for a full wrap. Mine is sewn to the esophagus - different from Joy's, but very similar. I could burp a little, and mine has stayed intact even though I had a bad episode of attempted vomiting (medication I didn't tolerate) which caused a reherniation. I don't have reflux, but I'll need a fix of the hernia.
A normal person, particularly men, with a good swallow should be able to have a full wrap. A lot of people with full wrap can still burp. No matter the wrap, most of us can't vomit.
Let the doc tell you what you NEED!