In my experience, you have a rare Suntour Perfect freewheel with the 13T small cog. I've personally handled 100s of Perfects and have had just one 13T Perfect.
The reason I mention this is that
if the 13T is worn or bad, it might be almost impossible to find a replacement cog. But it is my guess that your 13T is fine, as has already been mentioned, it's having too many links in the chain. The challenge is that if you shorten the chain to work on the 13T it probably won't play nice with the 34T while using the medium cage Shimano 600.
So another option might be to replace the 34T cog with a 32T or a 30T.