He is in a bad way, though I am not sure exactly what. It may have been injured during transport, this can easily happen from fish nets. Aside from the pale colour, there appears to be a red streaking near the front on the silver area, which would be blood under the skin. Diseases can do this, as can water issues, or injury.
There is no point in adding medication since we don't know the issue and with characins and corys any medication is going to be stressful so it should not be added without very good reason. I would just keep an eye on this neon, but I wouldn't hold out hope of a recovery. This is more than just mere stress.
I would also lower the temperature a bit. Both neons and the panda cory are cooler water fish than many tropicals, and 75-76F will suit them better; at higher temperatures, the fish is having to work harder just to live, and that is also stressful. The Betta will be OK at this lower temp.
On the Betta, have you seen any signs of aggression toward any of the neons? I honestly don't think that is the issue here, but if it is present it would add stress which makes any other issues even worse.