So I finally got my doghouse removed to put more insulation underneath.
Doghouse on the Axis:
Removed doghouse:
And I find that the dog house has about 1/2" of insulation all around it.
Even underneath there was insulation:
If you look carefully you can see a red arrow on the right pointing to the oil filler tube, and a blue arrow at the top pointing to the engine computer module.
(Just to identify some things.)
Since I already had purchased some material I doubled up the insulation (adding another 3/4" or so on top of the 1/2"). I checked the sound levels at 2000 RPM with a sound meter app on my phone and it didn't notice any difference (reading about 100 db before and after).