http://tateno-shika.com/general/perception/001.php
https://moriya-shika.jp/blog/歯が「キーン！」とすることってありませんか？/
https://www.morii-dc.com/blog/浩太先生のお口の疑問解決講座/666/

Teeth as well.

@@ -27,2 +27 @@
-<s_inf>as 頭がキーンとする, 頭がキーンとなる, etc.</s_inf>
-<gloss>with a stinging pain (from an ice-cream headache)</gloss>
+<gloss>with a sharp pain (in the head or teeth after eating or drinking something cold)</gloss>

daijs, prog, luminous

Not a noun.

@@ -14,4 +14,7 @@
-<pos>&n;</pos>
-<gloss>screech</gloss>
-<gloss>whine</gloss>
-<gloss>high-pitched sound</gloss>
+<pos>&adv;</pos>
+<pos>&adv-to;</pos>
+<misc>&on-mim;</misc>
+<gloss>with a screech</gloss>
+<gloss>with a whine</gloss>
+<gloss>with a shrill</gloss>
+<gloss>with a ringing (in the ears)</gloss>
@@ -20 +23,2 @@
-<pos>&n;</pos>
+<pos>&adv;</pos>
+<pos>&adv-to;</pos>
@@ -22,2 +26,4 @@
-<gloss>brain freeze headache</gloss>
-<gloss>ice-cream headache</gloss>
+<misc>&on-mim;</misc>
+<s_inf>as 頭がキーンとする, 頭がキーンとなる, etc.</s_inf>
+<gloss>with a stinging pain (from an ice-cream headache)</gloss>
+<gloss>getting a brain freeze</gloss>

きーん	21954  - GG5, ルミナス
キーン	161305 - Daijs   (many false positives from names)
きいん	8818 - Daijs
頭がキーン	4075
頭キーン	877
頭がきーん	270
頭きーん	34

This needs to be 3 entries. I'll split キーン, etc. out here and propose the others.

@@ -4,9 +4,6 @@
-<k_ele>
-<keb>キーン</keb>
-</k_ele>
-<k_ele>
-<keb>頭がキーン</keb>
-</k_ele>
-<k_ele>
-<keb>頭キーン</keb>
-</k_ele>
+<r_ele>
+<reb>キーン</reb>
+</r_ele>
+<r_ele>
+<reb>きーん</reb>
+</r_ele>
@@ -16,6 +13,6 @@
-<r_ele>
-<reb>あたまがきいん</reb>
-</r_ele>
-<r_ele>
-<reb>あたまきいん</reb>
-</r_ele>
+<sense>
+<pos>&n;</pos>
+<gloss>screech</gloss>
+<gloss>whine</gloss>
+<gloss>high-pitched sound</gloss>
+</sense>
@@ -25 +22 @@
-<misc>&col;</misc>
+<gloss>brain freeze headache</gloss>
@@ -27,2 +23,0 @@
-<gloss>brain freeze</gloss>
-<gloss>cold-stimulus headache</gloss>

For "キーン":

* From デジタル大辞泉 via:
https://www.weblio.jp/content/キーン
https://dictionary.goo.ne.jp/word/きいん/#jn-300441
See definition #2: "冷たいものを急いで食べたり飲んだりしたときに、頭に痛みを感じるさま。→アイスクリーム頭痛"
* https://www.glico.com/jp/health/contents/bokujoshibori12/
* https://m.facebook.com/nihongoflashcards/posts/2229484497295753/

--------------------------------------
For "頭がキーン":

* https://peta-eri.com/brain-freeze
* https://www.wanikani.com/vocabulary/井戸 (see parallel translation in "Context")

--------------------------------------
For "頭キーン":

* https://www.yorimichi-life.com/ice-cream-headache/
* https://englishnotes.jp/brain-freeze/ (form without が in the examples and summary definition.)* 
* https://www.youtube.com/results?search_query=頭キーン (e.g. https://www.youtube.com/watch?v=EJhM5NksKmk (which goes into an explanation of the phenomenon), or https://www.youtube.com/watch?v=uFeh708NRaA which provides a parallel English translation in the title)
* https://twitter.com/search?q=頭キーン&src=typed_query

I can't find much by way of solid established references on the phrasal versions here, especially without "が" as I encountered it, but I came across this term while reading, and some basic searches among colloquial sources (YouTube, Twitter, etc) and some study guides seem to seem to bring up solid evidence for the meaning here with all three readings I'm including.

It seems pretty clear that "頭キーン" comes from "頭がキーン" with "キーン" itself having a good definition in the デジタル大辞泉 (see references) pointing towards "アイスクリーム頭痛" / "ice cream headache"  as a synonym. It calls the part of speech as an adverb though, which seems odd for the headache itself, which I'd think was noun (as "アイスクリーム頭痛" is listed in EDICT, or "ice cream headache" is in English), so I listed it here as such.