Kata

There are too many cases to catch by random tests. Currently you solution is correct, but the other 3 (including mine) are not. It looks like one of those problems where all combinations can and should be tested, something like:

describe("all combinations", () => {
  const alphabet = "АБВГДЕЁЖЗИЙКЛМНОПРСТУФХЦЧШЩЪЫЬЭЮЯабвгдеёжзийклмнопрстуфхцчшщъыьэюя -";
  const tests = [];
  for (let x of alphabet) {
    tests.push(x);
    for (let y of alphabet)
      tests.push(x + y);
  }
  for (const s of tests) {
    const expected = referenceImplementation(s), actual = romanize(s).replace(/ʼ/g, '’');
    if (actual != expected) {
      Test.assertEquals(actual, expected, `Input: ${s}, expected: ${expected}, actual: ${actual}`);
      return;
    }
  }
  Test.expect(true);
});

(assertEquals only for failures to avoid too much output.)

You should specify precisely what the Cyrillic vowels are, it's not obvious to an outsider looking at the alphabet.

It looks like ʼ should be preferred in this case: https://en.wikipedia.org/wiki/Apostrophe#Unicode

U+2019 ’ RIGHT SINGLE QUOTATION MARK Punctuation apostrophe. Serves as both an apostrophe and closing single quotation mark. This is the preferred character to use for apostrophe according to the Unicode standard.

U+02BC ʼ MODIFIER LETTER APOSTROPHE Modifier letters in Unicode generally are considered part of a word, this is preferred when the apostrophe is considered as a letter in its own right, rather than punctuation that separates letters. Thus this letter apostrophe may be used, for example, in the transliteration of... the Cyrillic "soft sign"... Some consider, though, that this character should be used for the apostrophe in English instead of U+0027 or U+2019. Also ICANN considers U+02BC as a proper character for Ukrainian apostrophe within IDNs...