Provided you use sufficient preheat and interpass temperature, you should be able to weld without cracking. The postweld heat treatment above the critical temperature, i.e.,, lower A1, will serve to relieve any residual stresses and if quenched, will require a tempering operation to temper the Martensite present after quenching.
The PWHT should be performed immediately after the welding. Cracking, if it occurs, would most likely occur within the first few hours after cooling down to ambient temperatures.
You may pick up a little carbon through dilution with the 4340, but the dilution will be minimal using GTAW. Again, sufficient preheat is the silver bullet that will make this work.
Best regards - Al