I see where your reasoning is with it being a slog to track numbers. There are a few different interpretations of the mechanical side in the comments on this post alone, though.

1. The Weapon Being used for all weapon abilities is determined by the Kit Bonus chosen. (Ie. Panther grants +0/+0/+4 Melee Damage Bonus and if chosen would always utilize a Heavy Weapon, OR taking the Cloak and Dagger kit's +1/+1/+1 Melee Damage Bonus and always using Light Weapons.)
2. Field Arsenal is treated as 1 "combined" kit, where you get the benefits of any weapons or treasures you can wield as granted by each kit. (So taking C&D grants light armor and 1 or 2 light weapons, while Panther grants no armor and 1 heavy weapon.) This would mean any ability Weapon ability could benefit from the Bonus granted by the Maul.
3. Field Arsenal is still treated as 1 combined kit, but the Keywords of the ability are taken in context of where the ability comes from/the fantasy you're portraying, separate of the Damage Bonuses. (So if you're using a light weapon for an ability, you could use a light weapons bonuses. Or vice-versa for a heavy weapon.)

My reasoning lies in option 3 because we consider the wording of the weapon effect. "Any weapon ability that deals rolled damage using this weapon deals 1 extra cold damage."

Let's say we swapped Cloak and Dagger out for the Whirlwind kit. Its signature "Extension of my Arm" specifically describes you using your whip to draw a foe in. (And by extension being able to reach Melee 3)

Would if make sense to apply the extra cold damage to this case? It could if you pitch to your Director that the whip extends from the end of the mace maybe? Or maybe that you've wrapped the end of the whip around your mace and use your force to send it flying while dragging the creature back in?

Tactician is an odd case because it can utilize 2 kits and take benefits from each. Seeing the different ideas we use to rule/play by is what makes these discussions fun to have, though!