Hi, MKing.
Some time ago, I had to swap out the instrument cluster on my 05 (parasitic draw issue killing the batteries- I posted about it). Understand you already got the issue solved and thankful it was simple, but for the record I can confirm like Prarie Dog, the keyless entry module on mine seems same- it's a black box about the size of a tic-tac box, plugs into rear of instrument cluster. Not sure if the 06 would be (have been) same, but glad you don't have to worry about it.
Two quick side stories, if I may, one might save a fellow forum reader headache:
[1] Back when I did that cluster swap, BTW the new cluster from the dealer was hideously expensive... . The dealer would not allow both old and new clusters to be in my hands at same time. Dunno why. I had to remove my old cluster and pass it over the parts counter, in order for them to give me the new one. The parts guy did not know about looking for the keyless entry module. It was only because I took a very close look at both clusters to make sure the new one looked like it would fit. I noticed the new cluster had an empty bay that looked like something plugged in, where the old cluster that bay was occupied by a black tic-tac box. (honestly I'm not even sure I knew it was the keyless module at the time). I just saw the difference and said to the parts guy- maybe we should pry this thingy out of my old cluster and put it there in the new one, eh ? At least that's how I remember the story now...
[2] Separate but maybe related, on the topic of "replace your key fob batteries"... .
I bought a used 2004 VW Jetta recently. I was very fortunate to receive with the car both original key fobs, as well as the original valet key. It didn't take long to notice that the doors would lock with the key fob, but not unlock. I had to stick the key in the door to unlock. This was consistent, I tried several times to unlock with the fob- no unlock. Then I tried the other fob- locked and unlocked fine. So I replaced the battery in the first fob, and now it locks and unlocks. I'm not sure if the battery was low, or maybe I disturbed something with the fob unlock button when I had it apart. But it works now. And there you have it...
Wish you all the best.