I don't know about Baltistan but Gilgitis definitely hate Kashmiris (of Srinagar valley) probably for centuries, it is deep rooted hatred in them against Kashmiris of IOK. But for some strange reasons the souther parts of Gilgit regions, that is Astore is very Kashmiri-philic because of heavy influence of Kashmiri culture on them whereas regions west and north of Astore have always been Kashmiri-phobic. In old times Kashmir (Srinagar valley) was dominant kingdom of this area and ancestors of Dards (Gilgit, Chilas etc.) used to pay tribute to Kings of Srinagar valley for most part of their history. In fact according to some old histories of Kashmir, the Kashmiri royal army in ancient times often used to abduct "beautiful white girls" from the regions of Gilgit/Baltistan during their raids and bring them as booty back to Kashmir valley, conversely the Dards (Gilgitis) also used to make raids on the northern frontiers of Kashmir valley throughout the history. So people of Kashmir valley and that of Gilgit region had a rivalry since ancient times. No wonder Gilgitis hate Kashmiris and don't want to be part of any "Kashmiri identity"
. Another fact is that Shiaism that is practiced in Gilgit-Baltistan was spread there by "Kashmiri Shias" who were a failure back in Kashmir valley but proved to be very successful in northern lands.