One can establish "Siddur (Karaite)" on  the analogy of "Siddur (Hasidic)," already established.  "Karaite" is  the correct adjective
for the Karaites, as "Hasidic" is for the Hasidim.
Citing the entry from  EJ describing the Karaite sect in the NAR is  not quite appropriate for  the 670 field since  it  doesn't relate *directly* to  the heading being established (the siddur).  In order  to retain the information it provides without misusing the 670 coding, it can  be tagged as  a 675--  a source in  which the *heading* was  not found, but which is still relevant to the stablishment of the heading.

As  for  the  heading  "Karaites.  Liturgy  and  ritual"  (n 79-86364): this, like other "Liturgy and ritual" headings, is not valid under AACR2.

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If indeed the whole text of the Sidur is in Hebrew in Cyrillic script, then the uniform title will not be affected.  If the Sidur also has the text in vocalized Hebrew in Hebrew script,  that still doesn't affect the u.t.  One should have a 546 reading something like "Vocalized Hebrew and Hebrew in Cyrillic script; rubrics in Tajik; introd. also in Tajik," or else just "Hebrew in Cyrillic script; rubrics in Tajik; introd. also in Tajik"--whichever is the case.  (If there's no introd. that's NOT in Tajik, just drop the "also.")  The 041 will have indicator 0 and read hebtgk because of the rubrics (not because of the introd. or the text).

As for the classification, since there is no indication of any translation, the work would class with the original Hebrew text.
Since there was no mention of any specific rite, it would then class in BM674.2.