OK, esp. dropping the sense 3 from 痛い.

gg5, chujiten

All the kokugos and JEs have an entry for 甚く/いたく. I don't think it should be dropped.　Also, I don't think this いたく is just the adverbial form of sense 3 on our 痛い entry. They're defined differently in the kokugos and the 痛い sense is archaic.
I think this is a better approach. I suggest dropping sense 3 of the 痛い entry.

@@ -15,0 +16 @@
+<gloss>extremely</gloss>
@@ -16,0 +18,6 @@
+<gloss>deeply</gloss>
+</sense>
+<sense>
+<stagk>痛く</stagk>
+<pos>&adv;</pos>
+<gloss>painfully</gloss>

I suggest dropping this entry altogether and just having the rare/uk sense 3 of 痛い/甚い. 痛く almost always means "painfully" and if it's found in the wild, this entry will be matched. None of the EJs I checked gloss "exceedingly" as いたく.
There is one Tanaka sentence linked to this entry (私は彼のスピーチにいたく感動した) but it could be relinked to sense 3 of  the 痛い entry.

いたく感動	22893	0.5%
痛く感動	3643	0.1%
甚く感動	1027	0.0%

甚く感心	158	2.3%
痛く感心	427	6.2%
いたく感心	6284	91.5%

weblio even writes with kana
https://thesaurus.weblio.jp/content/いたく感動

One Japanese human being uses 甚, not 痛, to explain this sense (exceedingly), but most discourse is in hiragana
https://ja.hinative.com/questions/16673418

Wanted to bumping "甚く" for "correctness", but 痛く clearly dominates book results as well. Either way, [uk]. (Probably because people feel in their hearts that "exceedingly moved" should not be read as "painfully moved", but don't know the kanji to use instead).

(感動しました is the most common collocation with 甚く, followed by 感心)

Got here from the entry on 痛い, which already pairs these kanji.

@@ -3,0 +4,3 @@
+<k_ele>
+<keb>痛く</keb>
+</k_ele>
@@ -11,0 +15 @@
+<misc>&uk;</misc>