The 340/360 swap can be done in a 63-66 with some mods to the driver's side motor mount or a Schumacher kit, but you're into custom territory with the 60-62.
I would look into swapping in a '63-'66 V8 K-frame, and then modifying the driver's side motor mount as you would with a 340 or 360 into a '63-66. Exhaust will be an issue as well, as I don't know of anyone making headers for your specific application- maybe the later headers will work. Manifolds from a '72-74 would be your best bet if you aren't using a highly modified 360. The trans crossmember will have to be custom built. Use a V8 Valiant steering center link.
Obviously, the stock transmission won't work- use a 904/998/999 from a LA motor. Linkages will have to be fabricated for the shifter. (Don't get me started on the pushbutton shifter-if you want to keep it, either look for a used V8 cable shift tranny, and live with the ball and trunnion driveshaft, or contact Imperial Services
and see if they can help you out.)
Under no circumstance should EVER consider keeping the stock (7 1/4?) rear end- an A-body 8 3/4 is a MUST for this swap. (You can use a later and cheaper 8 1/4, but you will then need to upgrade the front bolt pattern to match the larger 5 on 4 1/2" bolt pattern the 8 1/4 will have) You also will need to upgrade the front brakes to something bigger than the tiny drums that thing has stock. You need a true dual circuit tandem brake system with a late model master cylinder.
A new driveshaft, much larger radiator, different throttle cable, and issues with clearance for the heater box will also need to be addressed.
I'm not saying it can't or hasn't been done, but it's no cakewalk. If you decide to do it, post up pix- I have a '64 Polara that I'm swapping a 440 into. (yeah, I'm keeping the pushbutton shifter in it)