Couple of points...
What is the size of the Area of Operations and does the size of the Area vary with the terrain? (Prairie-Desert-Arctic vs Jungle-Forest-Urban)
A long-range mortar, with lots of ammunition - doesn't that need a vehicle in support? Even a "dismounted" mortar benefits from an ATV to hump the ammo.
But.
While recognizing the value of large, lumbering, heavily armoured kit - I still tend to look at the value of light forces, even in a near-peer environment, as being to rapidly exploit opportunities to seize strategically important ground before the enemy gets there.
So, for me, the 60-81 mortars (with or without ATVs/UTVs and 2-7 km range) that can be dropped by 'chute or lifted by helo, make sense for light forces or even for rifle coy tms. On the other hand the 120 seems to me to demand a vehicle and if so it might as well be a variant of one in current service, so the LAV. And if the LAV then an automated, permanently mounted mortar.
As to the cap badge - that comes back to the intended employment an the area of operations. My sense is that a 20 km mortar is going to cover a large AO - an area considerably larger than even a 4 coy, 600 bayonet battalion could secure or invest, even in open terrain. That brings the debate, in my mind, down to Blackhats or Gunners, or perhaps both. A 20 km mortar attached to a Cavalry-Light Armoured Regiment makes eminent sense. Likewise one attached to an Infantry Brigade equally makes sense. I lean towards the notion of SP 120s as Arty tools that can be used to support both Cavalry and Infantry as occasions demand.
Perhaps the Brigade Arty should have a large SP-120 battery of something like 16 turrets in 4 tps of 4.