340 or 360?
A 360 is just a stroked 340 with larger main bearings, but a 360 can be bought for a whole lot less money than a 340.
Comparing stock stroke to stock stroke, at one hp per cubic inch, the 360's twenty extra cubes will be worth 20 extra horsepower at the same compression ratio. Back in the seventies, eighties and into the nineties, there weren't any good, inexpensive zero deck pistons for a 360, which hurt 360's compared to 340's, but that's no longer a problem.
The 340's smaller main bearings might cause a bit less friction loss than the larger ones in a 360. But I would think a 4-inch stroker crank might be more rigid with the larger 360 mains than with 340 mains.
At any rate, they are pretty close to being the same motor. I am running one 340, plus have an standard bore spare 340 block. Also running one 360, with one spare 360 block. My 340 and 360 are equipped pretty much identically - cam, intake, heads and exhaust - except the 340 has maybe 1/2 to 3/4 point higher CR. I can't really tell any difference between the two in the seat of the pants, although that's complicated a bit by the 340 being a 4-speed and the 360 being an automatic.
For a concourse restoration of a factory 340 car, you kinda need a 340, although once in the car the two motors can hardly be told apart, especially if the 360 is internally balanced. For a modified motor, it makes more sense to start with a 360 just because it's so much cheaper than a 340, and originality doesn't matter anyway.
Just my two cents.