I tried it several times. Doesn’t make any loud noises at all.
OK --- if you followed the directions very carefully - we have to assume that the output shaft from the transfer case is physically attached to the driveshaft and thus to at least the rear differential and from there ultimately to the ground. This is good.
NOW we have established the power flow from the parking pawl and ring through the transfer case.
I SUSPECT that we are on a wild fig hunt here (not a typo --- "fig" it is) ... and the next point of conjecture is inside the transmission ... sorry for that.
To recap --- that there are no really nasty sounds from trying to put the transmission into Park from idling in Neutral almost 100% proves that the integrity of the transfer case --- in that it is NOT in neutral - nor busted in some arcane way --- there'd then be the banshees from hell telling you otherwise.
It - and all it's associated parts - are sitting still, not spinning and that's good because if they were spinning, we go outward ---> toward the rear end.
As it is, we now turn the other way <--- and think inside the transmission.
That's because the results shift the problem into the transmission - somewhere before the parking pawl/hub assembly --- and this is where the money comes in.
Here's a copy/paste from an authority on this same subject:
1/ No movement in any gear: pump failure, or total loss of fluid.
2/ Remove transmission and rebuild, or refill pan and find the leak.
3/ If out of fluid, avoid running the engine until the transmission is refilled to avoid pump damage.
4/ To check for pump failure, check fluid level with the engine off, then start the engine and recheck fluid level.
If level does not go down when engine is running, the pump is broken.
Me here --->
I'mma gonna disagree with the whole concept of missing fluid --- I'm sure you know how to test the fluid level, so I'm not gonna insult you and ask if you've done that last part --- right? Right!
MY next step would be to run a pressure test of the pump at the tap right about where your right foot would wind up if there was no floor under your pedals .... it's a 1/8" NPT and you can "borrow" a gauge set from most parts houses for a nominal deposit which is returned when/if you return the tool.
Why do I go here? Because you kinda-really did indicate that you got wheelspin until the tires hooked up and then the transmission went into neutral and the vehicle hasn't moved or made any noises since.
That is never good - but we had to eliminate the transfer case and the driveline first - 'cause if they were broken, whatever quit would allow things to spin at a relatively high speed in Neutral and when you tried to put it back into Park --- the sound of the Parking Pawl trying to get into a notch would be something that nobody could mistake for a slight noise.
There MAY be some mitigating problem that only YOU can hear that you may not think is important, so I reserve the right to be a tiny bit incorrect.
But I think this is kinda cut-n-dried.