Sounds like a big no-no to me. You are supposed to have the same tire on each end of an axle. It is ok to have different tires on the other axle as long as they are the same at each end of that axle.
The C might be a trailer tire and the E a truck tire from the OP. You pick the psi of the truck tire from the weight it is carrying, while the trailer tire is usually at its max regardless of weight (from what I have gathered about all this)
IMO you have an unsafe situation there. Better to get a second new tire and maybe have the older C as a spare. If you do end up with the C on there after a flat, drive slowly to the garage and then put it back as the spare.