I used the generic classic auto air one for hotrods in my 65 coronet.
since mine was a non-ac car, nothing poked through the firewall.
I created a plate that bolted to the firewall and then bolted their a/c unit to that plate.
that way I didn't have to drill new holes in the firewall.
I looked at vintage air systems, and see they are very similar.
mostly I didn't have space issues because I could shift the airbox mount.
But I found I did screw up in that the classic box had forward facing vents for defrost and I didn't put any spacers in, so I was too close to the firewall to mount the tubing on that side.
I think they only thing that might be in your way is the fresh air vent on the passenger side.

I would measure how much room you have from any bracket holes for the upper area the dash mounts to just before the firewall curves to the floor for top to bottom. then where the fresh air vent is to where the ash tray is, assuming your 63 is similar to a 65.
they have changed there stuff a bunch since I bought mine, but I used their universal kit at the time. which looks like the street rod kit now.
The custom cooler looks like it would be easier to route the vent tubes, but no idea on the size difference.