The original flip sight should have been removed when the 5C is installed.
The 5C mounts using the pivot-hole of the original sight so I cannot see that you could mount it properly with the original sight in place - all you could do would be to use a bolt all the way thru, and dispense with the tilt (spring and ball bearing) mechanism.
The mounting lug on the 5C is (normally ?) threaded so you need to bolt thru from the 'inside' of the action.
If you have a bolt that goes all the way thru both pivot points you have to be very careful not to over tighten as you can pull the sides of the action in and trap the bolt (ask me how I know !!). The correct way to use a 'thru bolt' is to have a lock nut on the inside of the LH side pivot point to clamp the 5C against, (which means the original sight MUST be removed) or remove the original sight and use a short bolt thru the LH side pivot point only (which means the original sight MUST be removed)