The '63 heater box partially intruded into the engine bay, in '64 they were modified along with the firewall to accommodate the then-new 273 V8. This is why '63 A body V8 swaps need to have the offending section of heater box modified or eliminated.
What you need is a '64-'66 heater box; I don't believe the '63 version can be made to work in a '64 and up. Maybe someone knows a work-around on this, but I'm doubtful.
I'm starting to think the broken one is from a '65/ '66(?), as there was a slide-type dash control assembly in the box of parts that came with it. I'll try dry fitting the '63 box to my firewall before I go any further with it.