Operational needs
Virescit Vulnere Virtus
OK. You need to get in touch with your State Labor Board or equivalent immediately and report this. If FedEx can't pay it's bills...NOT YOUR PROBLEM. Nor, should anyone be forced to bring (and pay-out) personal funds for a clearly business-related expense. Tell management (in advance) that you are not paying a toll out of your pocket and that they can provide petty cash or a FedEx credit card for the fees. I doubt that FedEx would have any recourse if you refused to pay the toll or just sat outside the toll area waiting for a manager to come and pay it for you. In fact, that's probably what you should do. Come to work with just your DL and Medical card if you're DOT, and your lunch. Then, refuse to pay the toll, OR take the long alternative route you mentioned. I don't know if all of your couriers have this option.
I do know of a local RTD who refused to pay the toll on a bridge out of his own pocket and that he parked it until a manager came out and took care of it. 5 AMJs of P1, just sitting there for 90 minutes. Glorious.
You should already know that management is held to a much lower standard than you are, and that they regularly lie to cover-up their misdeeds, failures, and general incompetence.
AND, as someone else recently said on here. NEVER, EVER TRUST MANAGEMENT. They can, and will, toss you immediately under the bus to save their own skin(s), regardless of how great an employee you are or despite any friendships. Seen it happen many times.
That's AWESOME about the RTD though. I would have been dying laughing.
if I had to take a random Wednesday or something like that as an off day. There should be a law that, unless agreed upon by the employee, you should have two consecutive days off during the week. 
