Pag Island is well known for at least 4 things, the first of them you meet (or at least their parents) just after crossing the bridge! Paski lamb is consider to be one of the best lambs you can get in Croatia, if not even in Europe, do to the high concentration of salt on the Island, the sheep’s diet consisting of grass and wild herbs gives the lam from here a very special taste.
Driving would be a good alternative but maybe you are set on not renting a car. Could you possibly bus to Zadar, which should be more frequent from Split, and then find transfer from there out to Pag.