I think you may be mistaken about the general status of self-hater in English usage. The problem with running Ngram requests for hyphenated words is that the Ngram search program doesn't know what to make of hyphens (which, after all, appear frequently in textual material as end-of-line breaks between syllables of individual hyphenless words), so the results that Ngram returns are not (in this case) for "self-hater" but for "self – hater"—whatever that means.
In any event, a Google Books search for "self-hater" yields dozens of matches in published works going back to as early as 1850. For example, from an 1850 transaltion of Alessandro Verri, Roman Nights; or The Tomb of the Scipios:
This ever-during instinct is proofs against the cavils and dogmas invented, it would seem, for the purpose of driving man to utter desperation, or to make him a self-hater ; deeming himself a poor compound, neglected by heaven, and designed to return to dust.
From Thomas Moor, Counsels and Thoughts for the Spiritual Life of Believers (1882):
It is right in all things to avoid being a self-worshipper, but it is no less right to avoid being in this sense a self-hater.
From Margaret Oliphant, The Wizard's Son: A Novel (1884):
"At school, at home, abroad, in all your relations? Self-lover! My object at least is better than yours."
I am no self-lover ; rather self-hater, self-despiser."
"It is the same thing. Self before all. I offer you something better, the good of your race."
From Gerald Morgan, "Stevenson's Three Rogues," in The Yale Literary Magazine (March 1899):
There Herrick confessed to his impotency; confessed that he was weak, cowardly and a self-hater, and proved to the captain that he was no more fitted for accomplishing anything when right was against him than when it had been with him.
From Christopher Strong, "'There's Thou-sands in It!'," in The Green Book Magazine (July 1916):
This fellow was a water-drinker, and he laid it over all the hooch-hoisters I ever heard, even as Barnum & Bailey's lays it over the man on the corner dry-goods box selling Ogalalla Compound. Booze thickens the tongue of the self-hater and coarsens the work; but water, applied to the right organism, such as this under view, makes liquid and unending the discourse and, in the immortal words of Tennyson, flows around us coots forever, until it makes us haunt the fern.
And from Cornelius Hanford, General Claxton: A Novel (1917):
Knowing how others hated her, she became a self-hater. She was subdued and pitiable, as one who had purposely forfeited all right to the sustaining consolation of personal pride.
Altogether, a Google Books search for self-hater yields 110 unique confirmable matches scattered across 33 pages of search results.
The absence of self-hater from big dictionaries indicates that the word (1) isn't all that popular, and (2) has a meaning so clearly and fully derived from self-hate or self-hatred as not to require coverage in a separate dictionary entry. Because it is possible to generate hundreds of valid words by attaching the prefix self- to various nouns, dictionaries tend to limit themselves to covering only the most prominent combinations.
Obviously you are under no obligation to use self-hater if you dislike it. But I think that it is almost certainly the most common term in written English for someone who hates him- or herself.