If the car is rusty and you're inclined to bring the body back to it's former glory then you're taking it apart for whatever metal repair and body work it needs plus paint. If you can do that type of stuff yourself then you're somewhat ahead of the game but if you're paying a shop to do that work you could be over $20K before you even bat an eye. This holds true whether it's a small or big block.
If it's not rusty then well, how much would it take to make it drivable with the small block? You say it's been sitting so you're likely going to need to rewire the entire car, install new fluid lines, go through the running gear, brakes, steering and suspension, wheels and tires, interior, etc. That would also hold true whether it's a small block or big block.
Exhaust fitment is always a concern with these cars whether you stay with manifolds or go with headers. Front suspension uses different t-bars to hold the extra weight. Consider the intake you want to use and make sure you have hood clearance. Etc.
Like I said before, converting from a small block to a big block involves a little more then just motor mounts. Case in point; my friend has an original 318 '66 Fury which we are converting to a 383 based stroker (438") drivetrain. Even after literally giving him a bunch of parts from my shelves towards the project, (accessories, brackets, pulleys, ignition, sending units, intake and carb yada yada yada) he still has about $6-7K in the engine, $800 for a freshly rebuilt BB 727 and he still needs a converter. He's also into it for a couple thousand more for all the peripheral stuff like motor mounts, big block engine wiring harness, cooling system and radiator, fasteners, like $15-$1,600 for full TTI exhaust from the headers back and on and on.
This is strictly my opinion but unless you stroke the 383, I don't think it's anything special. The mostly stock 383 in my Coronet runs OK but it could be better and more cubes would certainly help. If I had to choose which engine platform to go with in a street driven A body I'd go with a small block.
Honestly, a small block in an A body is a better package for chassis weight distribution and handling whether it's for drag racing or street driving. If you decide to stay with the small block and you have the means, a 340 or 360 based stroker can easily be punched out to big block size and make similar power and torque.
OTOH, if it's strictly for drag racing, do whatever it takes to make it run the number you want and don't look back.
