If you downloaded the PDF to the PC before authorizing your Adobe ID on the PC, then the PDF will be locked to a sort of anonymous ID that ADE uses when not authorized to a 'proper' ID. In that case you would not be able to read the PDF on the PC any longer after authorizing ADE with your own ID, or read it anywhere else. Unfortunately, in that case I do not think there is any way to return to a previous anonymous ID and recover access to the document.
If the book was downloaded after authorizing your Adobe ID on the PC, it should be OK. I don't know if Bluefire suffers the same as ADE and sometimes gets authorizations confused. Try deregistering Bluefire from your ID and reregistering it.
So if the PDF is locked to the anonymous ID, there's absolutely nothing you can do to read the file on the Bluefire Reader app?
It looks like your ADE was "authorized without id" at the time of book download.
So your book has anonymous authorization credentials.
Later if you authorize ADE with Adobe id, it will not help as that particular book does not get associated with new id.
Book and authorization association happens only at the time of download.
To solve this, Please contact your library to provide you a fresh download link for the book.