gg5
両てんびんにかけ	122	
両てんびんに掛け	26

@@ -9,0 +10,4 @@
+<k_ele>
+<keb>両てんびんにかける</keb>
+<ke_inf>&sK;</ke_inf>
+</k_ele>
@@ -16 +19,0 @@
-<xref type="see" seq="1440150">天秤にかける・1</xref>
@@ -17,0 +21 @@
+<misc>&id;</misc>
@@ -19,0 +24 @@
+<gloss>to have two strings to one's bow</gloss>

JEs

@@ -20,0 +21 @@
+<gloss>to play a double game</gloss>

両天秤にかけ	1469	74.0%
両天秤に掛け	325	16.4%
を：
両天秤をかけ	167	8.4%
両天秤を掛け	25	1.3%


https://imidas.jp/idiom/detail/X-05-X-42-8-0003.html


天秤にかけ	29686	78.9%
天秤に掛け	7959	21.1%

Reorder kanji forms.  Removing the xref to the less-common form.  Also, imidas has this form, but not the other.

I'm suggesting a different xref to 天秤にかける[1], as it helps elucidate the meaning here, even though 天秤にかける[2] *is* this sense (and imidas also supports this).  The xref is certainly more common in aggregate, but I don't know how often it is sense [1] vs. [2]. (Usually I think "to try to have it both ways" has a nuance of failure.  It's not clear to me that this is the case in Japanese )

I suggest removing "to try to sit on two stools". Made me wonder if 天秤 was stool-related. It is not.  This seems like quite an odd English idiom to me, google turns up very little (if you search in quotes).  In fact, apparently a french quote:
https://quotefancy.com/quote/1376384/Fran-ois-Rabelais-One-falls-to-the-ground-in-trying-to-sit-on-two-stools

Doesn't seem to have much traction. Even "sit on two stools" returns zero real news matches (there is one example where a couple are sitting on two stools, one each).

Twitter gives 4 hits, and they are all:
"Having to sit on two stools because your butt can't fit on one."

I'm going to be proactive and remove this. If it exists at all, it's way less common than the Japanese structure.  It seems like a disservice to suggest that anyone should use it as a translation, and I don't think it offers much in understanding the Japanese.

@@ -5 +5 @@
-<keb>両天秤に掛ける</keb>
+<keb>両天秤にかける</keb>
@@ -8 +8 @@
-<keb>両天秤にかける</keb>
+<keb>両天秤に掛ける</keb>
@@ -16 +16,2 @@
-<xref type="see" seq="2130920">両天秤を掛ける</xref>
+<xref type="see" seq="1440150">天秤にかける・1</xref>
+<xref type="see" seq="1440150">天秤にかける・2</xref>
@@ -18 +18,0 @@
-<gloss>to try to sit on two stools</gloss>

@@ -15,0 +15,1 @@
+<pos>&v1;</pos>