Well, technically it only establishes that there exist Klingons that look like the discovery Klingons in an alternate universe, not that there aren't also Klingons that look like discovery Klingons in the prime universe. That episode also seems to establish that there are universes that are temporally offset from the prime universe, as the transformations also include turning the Klingons into an older kind of Klingon ("proto-Klingons"), so it could just as well be that the discovery Klingons are just an older variant of Klingons.

This idea is strengthened by the fact that Discovery and Strange New Worlds are clearly shown to take place in the same universe, with Discovery introducing the characters and ship that appear in Strange New Worlds, and Spock from Strange New Worlds reflecting on the events of Discovery. And SNW later shows TNG style Klingons, establishing that both styles (as well as presumably TOS style Klingons) coexist in the same universe. So I don't think the Lower Decks episode resolves anything about the Klingon-look at all, it was just a fun reference/gag.

It could be that the shows exist in different timelines, but it could also be that any of the shows exist in different timelines that are just similar enough to make the references make sense. For instance, Lower Decks establishes that a Picard exists in it's universe, but also exists in many other universes (see the purple enterprise), so it doesn't necessarily take place in the same universe as TNG.