R1b-Z2103 was the foremost Y-DNA haplogroup of the Yamnaya culture (at least its core settlements), it was already found even earlier in the Late Khvalynsk culture. So, I doubt its origin is Balkanic, or at least the specific group of Z2103 males that expanded to several other areas.
As for the thread's main question, I find it very unlikely that the Hittites and other Anatolian IE speakers of the Middle-Late Bronze Age, with the cultures and states they are associated with (I stress that because I do not mean the Proto-Anatolian Indo-Europeans and other intermediary populations), belonged to just one or two major haplogroups. I also do not believe they were still mostly of Proto-Anatolian origin, quite on the contrary, I think the evidences point to a linguistic shift caused by elite dominance. So, I expect them to have a lot of J2a, perhaps some G2a, J1, G2b, J2B, R1b, even L (considering it was present in the neighbnoring Transcaucasia in the Early BA). I don't think their early ancestors replaced the local paternal lineages much, instead they absorbed many populations along the way and were probably not mostly "PIE-like" when they consolidated their Anatolian IE languages.
Now, if you ask me what I think the major Y-DNA lineages of the Proto-Anatolian IE speakers were, since I still think it's more likely that they were somehow related to the Suvorovo-Novodanilovka expansion into the Balkans, I presume mostly R1b-M269 (Z2103? Probably) as well as I2 and G2 absorbed from EEF people... but anyone who's honest will tell you that we all just don't know, we can just conjecture. I think you may be onto something with the R1b-PF7562 assumption, it may even be ultimately wrong, but it sounds at least plausible.