We don't need every single agency to handle investigating threats to themself. It just doesn't make financial or practical sense. We can use the FBI, Secret Service, DHS, etc to deal with protecting the entire government and their employees. This will eliminate duplication and make it easier to share threat information.
One problem we have is the lack of information being shared with the correct agencies. This was allegedly the cause of some terrorist attacks not being stopped. If there are only a couple agencies that handle security and threat investigations instead of 100s it will already be in the hands of the correct agency. We should be striving for a streamlined process that removes duplication instead of hoping every single agency can do the job well.
Should every government agency have a department to investigate threats towards their employees? Why limit it to just the USPS? Why not give investigative powers to the Agency for Global Media or the Administration for Community Living? Should those employees have to risk the very "real and non-trivial" threats they face?
If this was just some sort of way of detecting bombs or anthrax or something I think most of us could get behind it. This is turning the post office into an investigative crime solving agency and not even strictly for the thing they do (mail delivery).