When McHale and the crew see the wallet in the alligator's open mouth the open end of the wallet is facing to the front but when McHale stoops down to try to get it the open end is facing to the side.
Disregard the alligator's simulated locomotion (rolling forward while it's feet barely touch the ground if at all - children's toys do this) and consider what an actual alligator can't do. It couldn't have walked up the steps to the hut where Parker was and it definitely couldn't have gotten on a chair at such a sharp angle trying to get at Binghamton.
The serial number he reads off the bill is way too short. (this may have been done on purpose to avoid any issues the Treasury Dept. may have had with reading off an actual number)
When Sydney goes to take Binghamton's wallet, you can see the cable operating him, behind his tail.