Hello and welcome to JustAnswer!
I have to wonder if the problem is with the body control module (BCM).
The ignition switch is wired to the BCM. When you turn the key to the crank position the BCM will then send a signal over the communication bus to the powertrain control module (PCM), the PCM looks at shifter position from the transmission controller and will then turn on the starter relay.
Since you're hearing the key-in chime when you shouldn't be and then it will crank as soon as that stops then that makes me suspect the problem is with the BCM. To know for sure what is happening the problem would really need to be diagnosed at the time it is acting up. A scan tool can be used to look at switch position in the BCM and whether or not the PCM is seeing the crank request. If the BCM isn't seeing the key go to the crank position and the switch is new then the BCM is the issue, provided the switch wiring tests ok. If the BCM sees the switch input change but doesn't send the signal to the PCM then again the problem is in the BCM.
Is there any possiblilty that this is the antitheft device and if so how would I know for sure.