I believe the proper installation procedure for the brake light switch is to depress the brake pedal and then install and rotate / lock the switch in the mounting bracket. Doing this prevents the EXTENDED plunger from contacting the brake pedal arm. Once the switch is locked into place, you release the brake pedal arm, the arm contacts the plunger pad and compresses it into the switch. The first movement of the plunger sets and adjusts its proper position. So I believe it is installed correctly.
Another simple test. When the brake pedal is depressed does the center, high mounted brake lamp illuminate and when released the high mounted brake lamp turns off?
I will give you a link to a dropbox folder that has 3 wiring schematics. Open the wiring diagram labeled fuses TIPM. Look under the hood at the TIPM and verify that the fuse in cavity M1 (rear center brake lamp) is good and has correct amp fuse. Likewise look at cavity 37 (antilock brakes, stability control, stop lamp) and it is proper and not blown.
It is tedious but I would check all the fuses and relays in the TIPM. If you look at the legend description, some fuses could be powering modules that might have some indirect influence on the diagnostic codes you are getting. The items labeled J are cartridge fuses and those labeled K are relays. Wiggle and make sure everything is seated properly and spade connectors are clean / not corroded. It is probably wishful thinking but sometimes in diagnosing problems the obvious approach gets overlooked.
Dropbox - 2011 Grand Caravan (at https://www.dropbox.com/sh/0q60pr2wn9c364k/AABzCUwC7I_FhcyZF_mbramWa?dl=0 )
If you get to this point and the center high mounted brake lamp passes its test and all of the fuses and relays in the TIPM are satisfactory, I would remove the brake light switch and do some testing. Here is a link to a video that shows how to test the terminal of the brake light switch. Perform the test and determine if the brake light switch is still functioning properly. You will need a 12 volt powered test lamp or a volt-ohm meter that can check continuity in circuits.
View: https://www.youtube.com/watch?v=nS1sUtgqcAs
In addition to the test shown in the video I would add this test. Seeing the orientation of the switch in the video with the switch plunger pointing to the left, mentally count and label the spade terminals from left to right in this fashion; 6 - 5 - 4 -3 - 2 - 1. With the plunger extended connect one lead of the test light / volt-ohm meter to terminal 6. In turn touch the other lead to terminal 4 , 3, 2, 1 in order. There should be NO continuity. Move the test lead to terminal 5. In turn touch the other lead to terminal 4, 3, 2, 1 in order. There should be NO continuity.
Repeat the above test with the plunger compressed. Press the plunger about 1/2 inch in. Use tape to hold the plunger into the switch assembly. Again touch a test light / meter lead to terminals 5 and 6 and in turn touch the other lead against terminals 4, 3, 2, 1 . There should be NO continuity. If the switch passes all these tests it is good. Installing an OEM switch is not going to fix this ongoing problem with code P0572 as you have tested and proven that the brake light switch is functioning properly. (I do NOT want to get into a discussion about the pros and cons of OEM parts versus aftermarket parts.)
If you get to this point and all testing still has not pinpointed a specific problem, then one is going to have to check wiring for open circuits, shorts to voltage or shorts to ground. In the dropbox link I have a document labeled P0572 diagnosis. It has a table of possible causes for code P0572. You would follow the diagnostic suggestions along with the diagram in document Brake Switch Wiring to PCM and trace the circuits.
Look at diagram Brake Switch Wiring to PCM. When the brake pedal is depressed the brake switch plunger extends and continuity is created between terminals 5 and 6 on the brake light switch. Terminal 5 is connected to ground and brake signal 1 wire is grounded. The PCM detects this and sends a message over the vehicle network to the TIPS to illuminate / turn on the rear brake lamps. When the pedal is released, the plunger is depressed and the circuit between terminals 5 and 6 is interrupted. The PCM detects this and tells the TIPM to cancel brake lamp illumination. However there is some intermittent condition (???) that is grounding signal wire 1 to the PCM on the dark green / white wire. The P0572 code indicates that the circuit from terminal 6 of the brake light switch is continually grounded and does not change state from OFF to ON and from OFF to ON. The PCM is getting logic confusion and thus does not send the proper signal to the wireless control module to unlock the shifter lever when the pedal is depressed and you attempt to move the shifter lever from PARK position. Shutting of the engine and then restarting temporarily clears the logic confusion or resets and the PCM sends the proper signal to the wireless control module to unlock the shifter when desired.
This is a lot to digest and the last part is my best guess as to theory of operation. View the video and perform all the brake light switch tests and then report back. Getting access to wiring and checking for shorts is a lot more involved.