Is the serpentine belt/tensioner pulley bouncing a bunch? If so, I chased this similar problem for several weeks back when I first got my Jeep CRD.
I changed the alternator pulley first, and it helped a little, then I changed the serpentine belt tensioner & pulley and that stopped almost all of the bouncing and 80% of the noise.
Finely, I replaced both motor mounts and that was the icing on the cake. All the noise at idle was GONE! Idles smooth as silk now with no noise!
Turned out the left motor mount was very bad and after I got it off if you shook it by hand, it rattled excessively internally.
Was simply a process of elimination......

Good luck!

Hard to tell watching video when ac is being turned on or off etc. but it looks like the alternator decoupler pulley has some play. The alternator pulley is a much more common part to wear out and could be making more noise with more resistant to serpentine belt rotation by engaging the ac. It usually leaves some characteristic red dust around it. Either way it would help to remove tension on the serpentine and see if the pulleys have any play or make any funny noises when rotating .

If it turns out to be the alternator there is a lifetime guarantee one at advance auto for about $115 with coupons etc. The decoupler can be changed but it requires a tool and good decoupler pulley used to be about $30 .

Way back when I first had mine changed; I took it to a local starter/generator repair shop and he stuck an air impact wrench on it and bumped it once with the air impact and it came right off.
He used no special tools and told me some Toyotas also had this same override clutch on their alternators and he was well versed in changing them.
He screwed the new Litens Alternator Pulley on by hand and bumped it one time with the air impact and handed me the alternator.
He did not even charge me any labor for changing it out.

So, when my alternator died about a year later, I used the same procedure in my shop using a air impact to remove the fairly new Litens Pulley before turning the core into Autozone when I purchased a new alternator that came with a new pulley.
Saved the Litens Pulley for a spare in case I ever need it.

Way back when I first had mine changed; I took it to a local starter/generator repair shop and he stuck an air impact wrench on it and bumped it once with the air impact and it came right off.
He used no special tools and told me some Toyotas also had this same override clutch on their alternators and he was well versed in changing them.
He screwed the new Litens Alternator Pulley on by hand and bumped it one time with the air impact and handed me the alternator.
He did not even charge me any labor for changing it out.

So, when my alternator died about a year later, I used the same procedure in my shop using a air impact to remove the fairly new Litens Pulley before turning the core into Autozone when I purchased a new alternator that came with a new pulley.
Saved the Litens Pulley for a spare in case I ever need it.

Could you give us a bit more detail on this removal procedure please!
Do you drive the female Hex section in the clockwise direction when looking at it from the front or counter-clockwise?
Or do you simply drive the square 10mm female slot with an air impact and if so in which direction?
I am struggling to find the procedure on ausjeepoffroad that showed them using a hollow long nut that held the pully in place with a spanner the same size as the long nut and then drove the alternator shaft off of the stationary pulley using an air impact and somehow driving it through the hollow long nut.
I may need to order a Litens Coupler soon as the local Stealers charge a fortune and do not know what they are doing. After much research I assertained that my 2002 Export 2.5 CRD can use the same updated coupler PN as used on the 2005/2006 CRDS and that this is effectively a grease-filled Litens!
Edit to add: The link I was looking for was not on ausjeepoffroad but right here on LOST in the CRD section. User mrkake posted the procedure on Wed Jul 03, 2013.

Glad you got it done despite the aggravation . Did you determine what was making the noise?

the chatter is almost gone. but I think i'm going to replace the adp next. its rusty. there was some longitudinal cracking and seemed stretched compared to the new one.