It's not just animated clothing, other clothing pieces do this glitch as well, I was testing all the hats I've owned, and found that the following hats caused this glitch.
-Fancy Top Hat
-Royal Crown
-Backwards Cap
-Drivers Cap
Oddly enough, despite being animated, this glitch did not happen while wearing the Discoball head.
I have found out when equipping a hat, the spyglass will go between the legs of a character, and that removing a hat will cause the spyglass will fly away.
I had a similar issue with the "can't connect to the market place" thing, how I got it fixed was by archiving then reinstalling Minecraft.