It is possible to convert your existing pull-down door handle to a push-button. If you do, you will need to fit the push button handle as well as the reinforcing plate behind the door handle to prevent damage to the door skin.
The usual reproduction parts suppliers listed here on the SP can supply you with most of the parts you need for the conversion. I have never seen the reinforcing plate in a catalog – you may have to get that out of a junkyard.
We did this on our '54 on one door, which was transplanted from an older truck. We used fiberglass body filler on the holes left behind by the turn-down handle and door lock.
Off hand, here are some of the suppliers/numbers that you might want to contact, there are likely many more listed on the links section of this site:
Jim Carter: Door Handles - DP 103 $19.95 each
Chevy Duty: Door Handles -18-107 approx $40.00
Truck Shop: Door Handles - 52-01913 approx $40.00
Classic Industries: Door Handles - CX 1291 $49.95