Hink,
Nice find again. I may have to move to Michigan to start locating more good stuff...
I believe the Standing Bear Logo with that circular text first appeared in print in 1951.
If I remember correctly, the Standing Bear Logo with that circular text IN COMBINATION with the BEAR Script Logo first appeared in 1954 in documents, then later both appeared as silk screens on bows in what we collectors call the 1956 models, but actually some were made in mid 1955.
As I'm certain you noticed, the silk screen on your bow stringer is much larger than the silk screen on the bows.
The only Bear Item that comes to mind with perhaps one of two silk screen that are approximately the same size, is the is the Bear Brochure Display which was made of wood. I believe that display dates to the late 1950s, but I'd have to check on that for sure, as my memory sure isn't what it used to be... I'm sure Bear made lots of different sizes of these two silk screens for various items.
You may have one of these wooden displays with the two larger silk screens than used on the bows. Note it has a different color combination than the logo on your stringer...
The script Bear text is 3.25" long, the Standing Bear Logo is 2.5" to the outside of the text.