Kor, Kang, and Koloth went from TOS Klingons to TMP Klingons. So that doesn’t work.
No, it can still all be true.
Remember, we don't know HOW those Klingons went from QuchHa' to hemQuch. Did it assert itself naturally? Did they have cosmetic surgery? It's all possible.
If some aspiring writer wants to write an SNW episode featuring 'Final Reflection' Klingons, for example, they can do that if they
