It definitely sounds like a permission problem then. My guess is that you're execute the package in the SSIS Catalog via T-SQL (you didn't answer this question), and your Kerboros isn't permitting the double hopped authentication. As a result it's trying to authenicate as NT_Anonymous, which results in a failure. I would suggest checking with whoever maintains the network security that this type of hopping is permitted.

Have you also tried running the task via an Agent job? This only requires one hop on kerboros, so as long as you have it enabled, should work by default (provided the user Agent is running on has permission).