Will a .357Sig barrel made for the SP2340 fit with a .40 SP2022?

While there are slight variations due to manufacture it should drop in and fit just fine. The only difference in the Slide between a 357sig 2022 and 2340 is the actual the writing on the slide otherwise they are identical. The major change in design between the 2340 and 2022 was the rail on the frame.

Most vendors have the barrel listed for the Sig Pro as replacement barrel in 357Sig for Sig Pro 2022/2340.