Doubt it is the chain unless it's simply worn. Pretty much any chain works fine on a friction shifted UG freewheel. Like others, I prefer a SRAM PC 850 or 870 if one is at hand, but otherwise have used 9 or 10 speed chains from various manufacturers without any shifting issues. In my experience, a Shimano or Dura Ace freewheel body is oretty much a lifetime acquisition.

As mentioned above, I'd start by removing the wheel and wiggling the frewheel back and forth to see if the body is loose internally. If that's not the problem, and you're convinced that the hub itself is relatively concentric (mount a different feewheel to see if it also wonbles on that hub), then likely the cogs are simply worn. Unusual for the smallest cog or two to be the worn ones, (it's usually the 3rd or 4th), but it happens.

If you're going to be up here in the next few weeks, bring it along and we'll swap out the 13-15 to see if that resolves the problem. My guess is it will. If not, I have a closet full of 13-28's and 14-28's.