Carter 4441 fuel pumps that are original and date coded are valuable so rebuilding to make them reliable is a good option. I have found some U tube restoration videos do not reflect working on 56 year old parts, but this one at the 428Cobra Jet site is very good.
https://www.428cobrajet.org/how-to-fuel-pump I used this procedure successfully. Removing the diaphragm from the housing did require aggressive twisting to get it out. The old rubber diaphragm in my pump was cracked on one side. It had low miles but was in use since 1968.
On assembly, there is a stiff spring that was a challenge to install. Pictured is a method that you may prefer using a screwdriver to compress and slide the base of the spring in place. First though, there is no fulcrum (pivot point) for the screwdriver so you must create one by adding a piece of wood (3/8 inch wide) placed in the pump lever - see attached picture. Insert the screwdriver in the second coil from the bottom, compress the spring and push it into place. I also temporarily added a 1/4 inch bolt (7/8 inch long) underneath the spring to create an arch. See Pic
A well used pump body may require tumbling or vapor blasting. Dead Nuts sells an ink stamp for numbering. I was advised the pump is installed after the engine paint was applied. Finished pump shown. Good luck. Roger