yes, i agree. it's a negation

@@ -13 +12,0 @@
-<gloss>to the extent that ...</gloss>

"母の死を嘆(なげ)くひまもあらばこそ"=I did not even have time to mourn the death of my mother.

This one appears to fit with definition #2. It is being used for negation.

DAJG (Makino & Tsutsui), 中辞典

M&T comment that it can be replaced by からこそ, but not vice versa.
中辞典 has a second sense: "《反語》母の死を嘆(なげ)くひまもあらばこそ, 今度はリストラの憂き目にあった. Hardly had I laid my mother to rest, when I lost my job due to corporate restructuring."
Should this worked in? It doesn't seem to match the others.

@@ -8,0 +9,2 @@
+<pos>&conj;</pos>
+<xref type="see" seq="2700470">からこそ</xref>
@@ -10,0 +13 @@
+<gloss>to the extent that ...</gloss>

koj, daij, meikyo, prog, etc.

@@ -8,2 +8,3 @@
-<pos>&prt;</pos>
-<gloss>to be only because</gloss>
+<pos>&exp;</pos>
+<s_inf>after the -ba stem of a verb</s_inf>
+<gloss>only because ...</gloss>
@@ -12,3 +13,4 @@
-<pos>&prt;</pos>
-<s_inf>verb-negating suffix</s_inf>
-<gloss>not</gloss>
+<pos>&exp;</pos>
+<s_inf>after the -nai stem of a verb, usu. as 〜も...ばこそ, 〜など...ばこそ</s_inf>
+<gloss>definitely not</gloss>
+<gloss>certainly not</gloss>
@@ -17,2 +19,2 @@
-<pos>&prt;</pos>
-<gloss>if one actually</gloss>
+<pos>&exp;</pos>
+<gloss>if actually</gloss>

When I see "ばこそ" written in modern Japanese it is usually as definition #1. This definition is recorded in "Japanese: A Comprehensive Grammar" written by Stefan Kaiser and others, page 359. One example sentence given by this source is "周りの理解があったればこそですけだね。=It was only because I had the understanding of the people around me."

By contrast, in pre-1945 Japanese "ばこそ" usually seems to be some sort of negation. The Handbook of Japanese Grammar by Harold G. Henderson gives this example on page 57, "Utedomo tatakedomo ozuru mono no araba koso=Though I knocked and knocked, never a man would answer me!" A few week ago I read this similar example of "ばこそ" used for negation: "激昂せる兵は上官の制止をきかばこそ片はしより殺戮する。"

Definition #3 seems to be the literal combination of "こそ" with conditional tense "ば". The translation "if one actually" comes from The Handbook of Japanese Grammar by Harold G. Henderson (page 57).