GG5, etc.

Don't need all those straps. Trimming sense 2.

@@ -18 +17,0 @@
-<gloss>bill strap</gloss>
@@ -24 +23 @@
-<gloss g_type="expl">paper strap around a newspaper (or similar) with the recipient's mailing address</gloss>
+<gloss>mailing wrapper (around a newspaper, etc.)</gloss>

The wikipedia page on currency uses this as [1]:
https://ja.wikipedia.org/wiki/紙幣
帯封で束ねて札束にして整える

https://en.wikipedia.org/wiki/Currency_strap
A currency card, currency band, money band, banknote strap or bill strap is a simple paper device designed to hold a specific denomination and number of banknotes.[1] It can also refer to the bundle itself.[2]

Moved the wikipedia headword "currency strap" to the front.  Amazon results include many names, but "self-sealing currency straps" comes up first for me, and seems pretty reliable in search results.

Any evidence for [2] "rubber band"?  This feels like a few simplifications were made from "thing that wraps a newspaper", losing sight of what the original term means.

I don't think there is any such thing in English as a "newspaper band", at least not one that would be readily understood.

image search: 帯封 新聞
vs. image search: "newspaper band"

sankoku has [2], in detail.  This doesn't exist in English, so any gloss without an [expl] is going to be missing information.

note: sankoku says: 新聞など, so this isn't strictly limited to newspapers.
Another useful search: 帯封 郵便

One result is the postal service (with illustration):
https://www.post.japanpost.jp/int/service/pm_packing.html
新聞等:
帯封をします。

In the U.S., historically, newspapers were delivered by paperboys with lists of addresses (hence no 帯封).  In my experience, mail-delivered newspapers(etc.) have a sticker affixed directly to the paper with the delivery information. So, again, I don't think 帯封 can be simply glossed into English here.

@@ -15,0 +16,2 @@
+<gloss>currency strap</gloss>
+<gloss>banknote strap</gloss>
@@ -17 +18,0 @@
-<gloss>currency strap</gloss>
@@ -23,2 +24 @@
-<gloss>wrapper band</gloss>
-<gloss>newspaper band</gloss>
+<gloss g_type="expl">paper strap around a newspaper (or similar) with the recipient's mailing address</gloss>

gg5, jwiki, daijs

daijs has only the newspaper band sense but google image search suggests that money band sense is much more common.

@@ -16,2 +16,9 @@
-<gloss>half wrapper</gloss>
-<gloss>strip of paper</gloss>
+<gloss>bill strap</gloss>
+<gloss>currency strap</gloss>
+<gloss>currency band</gloss>
+<gloss>money band</gloss>
+</sense>
+<sense>
+<pos>&n;</pos>
+<gloss>wrapper band</gloss>
+<gloss>newspaper band</gloss>