Why don’t we say on hoshana rabah “vchatmeinu....” in the shemone esrei if it’s accepted that it is the real day of judgement at least according to Kabbalah?
Rav Yaakov Kamenetsky (in a much longer discussion of the status of Hoshana Rabbah and why Chazal don't explicitly say it that it is the final day of judgment etc.)explains that Hoshana Rabbah is similar to the concept of filing an appeal.
Yom Kippur is the Gemar Din. Therefore we say V'Chsmainu then. Hoshana Rabbah is the last day to change that judgment with an "appeal"but it still isn't a day of judgment for those not seeking an appeal Therefore V'Chasmunu would not apply.
Another reason is that at the time they made the Nusach HaTefillah, Hoshana Rabbah was not known as a day of judgement. The only reference in Chazal is a non explicit one in the Yerushalmi. (They did not want to say so explicitly so that Sukkos would not turn into an extension of the Eseras Yemei Teshuva ,Rav Yaakov Kamenetsky) Had they written V'Chsmainu in the Nusach HaTefillah it would not have been a secret anymore