It's true that "cut (of meat)" still means "part" of the body, but I still think it'd be more helpful to split it out into its own sense. 
Approving, though.

I agree that "part of a body" shouldn't be a separate sense. It just creates unnecessary overlap.
But I'm not sure "cut of meat" can be considered a separate sense either. It still means "part" or "region" in that context.
This is my suggestion.
---
I note that daijr's sense 1 ("position of a part") isn't even covered here. The JEs and daijs have ignored it as well, though.

@@ -16 +16 @@
-<gloss>part</gloss>
+<gloss>part (esp. of the body)</gloss>
@@ -19,4 +19 @@
-</sense>
-<sense>
-<pos>&n;</pos>
-<gloss>part of a body (e.g. cut of meat)</gloss>
+<gloss>cut (of meat)</gloss>

It's closer to the daijr definition for 
sure, but is it more helpful for our 
users? I think "part of a body" isn't 
really necessary to split out from sense 1 
from a wa-ei perspective, but "cut of 
meat" is.

I've been chewing over that daijr sense. How's this?

@@ -22 +22 @@
-<gloss>cut (of meat)</gloss>
+<gloss>part of a body (e.g. cut of meat)</gloss>

daijr does have two senses:
① 全体に対するある部分の位置。 ② 生物の体で、全体に対しての一部分をさしていうことば。 「身体各－の名称」 「食肉－」
daijr's sense two might be broader than just "cut" but considering compounds like 希少部位 I think "cut" definitely should be its own 
sense, esp. considering this is a wa-ei.