This certainly seems possible to me, without any special workarounds.
I created a folder in Google Drive for account A. The folder contained a PDF and a Google Doc.
I shared the folder with account B.
From account B, I found the folder in the "Shared with Me" section, right-clicked, and chose "Share". There was no restriction on me being able to share. In fact, I was able to share the folder with account C.
Back on account A, when I look at who the folder is shared with, I see both account B and account C.
So, it would seem that unless there's some key piece of information missing from your question, what you want to do is possible without any need for a workaround.
I went back and tried again. This time, in the "advanced" sharing options, I clicked "Prevent editors from changing access and adding new people".
Now, if account B tries to re-share, I get a note that sharing is restricted, and the ability to request that someone else be allowed access. Presumably this is what has happened in your case, and you don't want to ask for access for this other account you want to share to. (It seems that ethically the proper course is to ask for permission.)
Adding the folder to "My Drive" does not change the sharing restriction. There is no way within Google Drive to duplicate a folder in place. So, it would seem that no, there is no workaround other than the one you've already found: download and re-upload with a new name.