In this vehicle in particular, no, this is not really possible.
(I know this is an older question, but I've heard it IRL before so others may find it useful)
The PTS version will have an electronic steering lock, and (from diagrams I'm fairly certain) different BCMs (with different firmware or settings at the very least). Additionally, the immobilizer strategies used by the models are different. With proper bypassing of the immobilizer it may be possible to artificially replicate the 'turn-key' style by sending the right signals to the certification ECU (authenticates key data), BCM (likely needs to see a signal indicating the brake is being pressed), Steering-lock module (for obvious reasons), gauge cluster (on these models they're aware of the key system), etc, but it won't meaningfully change the operation of the vehicle.
More broadly, it might be more straightforward in some select vehicles, but the only ones I can speak to are Chrysler/Jeep/Dodge/Ram vehicles of a few years ago, but they're kinda backwards. In some years, on some of those vehicles the PTS or TIP-Key module can be switched out and operate in some capacity. The gotcha is that the tip-key versions are basically stripped down push-to-start systems, where the SKREEM module (the module controlling start/stop) is communicating with the rest of the vehicle in the same fashion whether turn-key or PTS. The vehicle still has automated cranking, anti-grind/overcrank protection, and it's still just a module sending (possibly encrypted) commands over a few wires to the modules actually powering up the circuits.
After years of installing remote start systems (basically automatically hotwiring/starting vehicles), I've never come across a PTS vehicle that could be started in the same fashion as its traditional-key counterpart.