This problem is caused by a combination of a lean fuel mixture, and either a spark plug design which is not effective at igniting lean mixtures, or weak ignition energy. Most mechanics would address this problem by making some kind of (illegal) modification to richen up the fuel mixture; but that is really not an appropriate solution, because it would raise the emissions and reduce fuel economy. If the fuel pressure regulator was defective; this could cause the problem. However, I expect that the $100+ fuel pressure regulator is really not bad; so I'd sure hate to bet my money that replacing it would make any difference, without testing it first.
This kind of problem is not uncommon in late model vehicles. It is often not caused by defective parts; but is instead the result of the increasing amount of oxygenating additives which are now being put into the reformulated fuel we use. The fuel mixture on many recent vehicles was originally calibrated very close to the leanest possible ratio, in order to meet that year's federal emission requirements. But as fuel becomes more heavily oxygenated; the mixture becomes leaner than it was originally. And this can throw some motors over the limit of cold idle stability; which becomes most critical in the first minute after the motor is started. Here's a simple and no cost solution to this problem; which I have found will very often solve the issue: When starting, turn the key to the position where the dashboard warning lights illuminate, and wait with the key in that position for ten full seconds. Then turn the key the rest of the way to start the motor.
The reason this works is that it takes about ten seconds for the electric fuel pump to build up normal pressure at the injectors. If you don't wait for full fuel pressure to build up before starting; the engine's fuel mixture will be excessively lean, so it will be more likely to stall or stumble. Putting the transmission in gear will place a greater load on the motor than idling in neutral or park; which makes it even more likely to stall when the mixture is too lean. Once the spark plugs come up to normal operating temperature (which takes about 1 or 2 minutes); they can ignite leaner fuel mixtures than they are able to when they have not yet warmed up. That's why the motor will not stall or stumble when it is started after having been run recently.
You also might have some success by using a brand of fuel which has a different additive formula in it. Shell is a likely brand to run better during starting, and Chevron is another.
If the simple approaches outlined above do not clear up the stalling; here's what I would do to deal with this problem: Start by testing the alternator for bad diodes and also confirm that it meets the manufacturer's minimum current output specification at idle speed. This is a more sensitive test than the usual alternator test, which just uses a voltmeter; so it should be done by an electrical specialist who uses an oscilloscope to test the alternator output waveform, in addition to a voltmeter and an ammeter. If the alternator is defective; replace it with a NAPA premium alternator. (I have encountered so many bad remanufactured alternators from other parts stores; that I only trust NAPA alternators anymore).
Check the auxiliary heater relay (also called intake air heater control relay) for excessive voltage drop and also check the current draw through the intake air heater circuit against manufacturer's specifications. You may need to check a Mitchell manual or a factory service manual to find that figure. Replace any defective components or repair any problems found in that system.
With the motor idling; compare the supply voltage at the ignition system or at the auxiliary heater relay with the voltage at the battery terminals. If the voltage at the ignition system or the auxiliary heater relay is more than 0.6 volts below the battery voltage; this indicates excessive resistance in the ignition switch contacts; so replace the ignition switch.
Have the fuel pressure between the regulator and the injectors (not directly from the pump) tested and compared to the manufacturer's specifications for THIS EXACT YEAR MODEL AND ENGINE SIZE. You may need to check a Mitchell manual or a factory service manual to find that figure, too. I would not trust the figures in the cheap consumer oriented manuals (Chiltons, Haynes) that you find at parts stores. If the fuel pressure is below the manufacturer's minimum limit, then replace the fuel pressure regulator.
In the unlikely event that the above measures have not eliminated the stalling; I would replace the spark plugs with a design which has a lower voltage requirement and also heats up more easily. Bosch makes such a plug for this motor. The best choice would be either a Bosch Platinum/Iridium Fusion plug # 4508, or the lower cost Bosch Platinum +2 # 4308. These plugs come with preset, non adjustable gaps; so do not attempt to adjust them.