You could but it's not a good idea. On Palaci the 105 mm is fired at charge 7 for a flat trajectory with a small beaten zone because the trigger areas are relatively small and the round can be thrown off by the high winds in a storm shoot (and many of the shoots are storm shoots) Winds would be a particular concern as the mortar round would have its max ordinate above the mountain peaks where the wind is the worst.
There are many parks around the world that do. Palaci only involves a small det (albeit four 6-week rotations) and is an excellent leadership development platform for junior officers and NCOs - especially for reservists.
I certainly wouldn't buy a new weapon. We have lots of C3s and could cannibalize parts for decades to come if we ever gave up the gun as a training gun. One thing is that there are in excess of a hundred target areas (can't remember the exact number) changing the gun would require reregistering all those targets which would be pricy.
My guess is that the only people who know about Palaci are the guys who are stopped on the highway during a shoot. They're usually bitching about the delay.
Theories differ, but that's basically my view based on the principle that it's the battalion's guaranteed indirect fire support system. Handing it to the artillery would risk having it become a brigade asset to facilitate massing fires and thereby no longer guaranteed to the battalion.