You can still stream the base64 separately and reference it inside the JSON somehow like an attachment. The base64 string is much more versatile.
Using base64 means that you must encode and decode it, but binary data directly means that is unnecessary. (This is true whether or not it is compressed (and/or encrypted); if it is compressed then you must decompress it, but that is independent of whether or not you must decode base64.)
There's nothing special about "text" or binary here. You can absolutely put binary inside other binary; you use a symbol that doesn't appear inside the binary, much like you do for text.
You use a divider, like " is for json, and a prearranged way to avoid that symbol from appearing inside the inner binary (the same approach that works for text works here).
What do you think a zip file is? They're not storing compressed binary data as text, I can tell you that.