I certainly haven't seen any original records, so you may be right that Alivia is a typo for Alicia. On the other hand, if it's a mistake, then it's a mistake that many people have made. Google "Alivia Braose" or "Alivia de Braose" and you'll find numerous geneological pages that reference her. Curiously, some of these websites also call her Aline, a name which is supposed to be related to Adeline, not Alice.
I looked for some more medieval Alivias and came across two more: both of them named Alivia Bassett, both living in the 13th century. The second Alivia Bassett is *also* called Aline in some sources. She was the mother of a famous English lord named Hugh le Despenser the Elder, and is referred to as Alivia by more than one source. So I suppose Alivia is somehow related to either Alice or Aline.
BTW, all my "research" comes from the internet, so there's always the possibility that there are some of the websites I've seen aren't too reliable. :)