According to the general regional pattern would China be called "China" in the South while "Kina" in the North, as well as the distribution of "ich" and "ik".
The general pattern you observe or quote is actually something different.
Your example being ich, which is pronouned ik in the northern dialects because these did not participate in the High German sound shift. For ich/ik, the boundary is the Uerdingen line which is sometimes used to separate Low German from Central German dialects. (The Benrath line or machen/maken line is also sometimes used.) The sound shift in question transformed the /k/ plosive of the North into a /x/ or /ç/ fricative in the South depending on the environment of the sound. (It also affected /t/ and /p/ in a similar manner.) Only in some Swiss, Austrian and southern German (Allgäu, Lechrain, Werdenfels) dialects all occurances of /k/ are affected; usually it did not apply to word-initials.
In practically all of Germany, most of Austria and probably some parts of Switzerland, a word-initial original Germanic /k/ remained /k/ but can be written either k or ch — the latter mainly in place names such as Chiemsee or Chemnitz. Other than those you won’t find any word-initial ch in non-borrowed words.
A few borrowings exist with word-initial ch from Italian, English or French which are pronounced according to the standards of the respective original language. The same can roughly be said for the borrowing Chuzpe and probably some other Eastern European ones that aren’t in my head right now.
Then there are a few words borrowed from Greek which have word-initial Ch because the original Greek letter was probably chi (I didn’t examine all etymologies). For some reason unknown to me these are pronounced much more as if they were originally German, just uncommon and thus weirdly spelt. Their pronunciation mostly follows the surroundings rule for word-internal and terminal ch: as /k/ if there is a dark vowel or consonant following (Chor, Chlor) but as /ç/ if there is a light vowel (e or i) following (China, Chemie).
For reasons not fully known to me, the southern dialects never adopted that rule and just stuck with initial /k/ for all of these sounds.
I haven’t been able to find a definite map of the extent of /ki:na/, but I can tell you that it is only a relatively small area in the South. There is a very broad band in the middle where it is both /iç/ (for ich) and /çi:na/ (for China).
It might be a matter of over-generalization. Just like everybody else, Bavarians and Austrians say /kɑ'rɑktə/, /krist/ and /klo:ə/ (Charakter, Christ, Chlor), but, over-generalizing (?): /ki:nɑ/.
However, the dictionary tells me that like all the others they pronounce Chimäre with /ç/, not with /k/. So it might also be a problem of orientation.