Here's one reader's quick response, for what it may be worth; a few points in no particular order and without the text of Coleridge's "Kubla Khan" at hand.
1.The poem exalts poetic geniuses (last few lines). The great oet is a man apart from the common folk. That attitude is completely in line with Dunsany's and Lovecraft's aestheticism.
2.The poem is about the loss of a glimpse of a realm of supernatural or preternatural beauty. Coleridge -- or "Coleridge," the speaker in the poem -- longs to see again what he glimpsed. This seems to be a concept that crops up in many fantasy stories. Without checking, I'd say Merritt's "Through the Dragon Glass" would probably be an example. (That story is in Lin Carter's 1969 anthology The Young Magicians, whose two Lovecraft stories, "The Quest of Iranon" and "The Cats of Ulthar," were probably the first HPL works that I ever read. The book was my intro to several other notable authors as well.) Would HPL's "The Silver Key" be another? I haven't read that one in ages!
3.Brief as it is, the poem adumbrates the familiar fantasy fiction and Lovecraftian knack of suggesting a depth that is not there explored. Lovecraft studded his stories with references to entities, calamities, and locales that he did not describe in detail, which Dunsany had done before him. (It's sort of like Doyle's/Dr. Watson's habit of alluding to "the giant rat of Sumatra" in the Sherlock Holmes stories, building up a sense of cases that there's no time or no occasion to relate now.) Coleridge, long before any of them, teases us with, for example, that bit about a woman "wailing for her demon lover" -- wailing because she long for him to come to her again? summoning him? wailing because she was deceived by a demon in the form of a man? We can't say for sure, but nor can we forget it.
4.Between the generation of Coleridge and the generation of the late 19th-century fantasists you have Poe. I think that, at his most fantastic, Poe is often writing, perhaps consciously, as a "Coleridgean." I'm sure that HPL would have brooded over Coleridge, but even if he had not, he would have received a strong indirect influence of STC via Poe. Poe's finest weird verse is Coleridgean.