The rear side plate screw appears to be a flat head type.
This was used with the Colt Target grips so the grips would lay flat on the frame.
Guns fitted with the Service type grips like those on your gun had round head rear side plate screws.
While this is NOT proof positive ( screws can be replaced), I suspect your 357 originally shipped with Target grips.
If so, the "correct" grips for that serial number 357 would be the First Type fully checkered Target grips with silver Colt medallions.
Unfortunately, these are now VERY expensive, usually starting around $200 to $300 and UP.