It's not the postage that would raise concerns for me -- it's the work flow.
I could, in theory, drop it in my departmental mail bag with the right bar code attached to it, and it would probably work. "Probably" isn't good enough for me, though, when somebody's career is involved. I would make a call or two to make sure it would work, and hopefully I'd get a valid answer.
In practice, with such a request, I would probably just walk the letters into a Post Office, and pay for it myself, just to make sure it would be handled appropriately. Assuming it's under $20US or so, it's a cost I would gladly absorb for peace of mind -- so long as the situation doesn't come up too often!