I'm not sure if many other people have tried teaming up with Big Daddy in 2vs2 but I have to say that Big Daddy is one of the best partners to have in 2vs2 for quite a lot of characters.
Much like in Bioshock, Big Daddy is a protector, he has a lot of tools to help keep characters safe or come charging in to their rescue should they be attacked. His Tornado and Ice are amazing for giving some extra cover for characters that suffer at close range and may be targetted, his armored attacks are good for simply knocking down or knocking away opponents who are attacking his partner and his various ways to stun, freeze, grab or juggle his opponent with down triangle makes him amazing to have for setting up other characters supers as well as his own. Long range fighters are pretty good for being able to give Big daddy cover at range, close range fighters can benefit from Big Daddies armored attacks and traps creating openings for combos, mid range fighters can use Big Daddy to either increase their close range capabilities using his close range attacks or increase their zoning thanks to Big Daddies plasmids.
I genuinly think the devs designed Big Daddy to reflect his role as a protector of the Little Sisters in the Bioshock games. If anyone hasn't tried using him with a partner or having him as one I highly recomend giving it a shot with any half decent Big Daddy.