Presumably the magic of the anti-magic field sets them off.
Dragon's Fireball: Make the Greater Fireball use the waste heat for additional power already.
Study Staff Material: Okay, sure, it's not ideal, but let's see what we can learn from the staff fragments. Maybe it's the material itself that has the anti-magic properties.
Micro Fireball: It's even smaller and shorter ranged than our Streamlined Fireball. Why make this? So we can have a go at teaching our normal forces a little magic. Even if each of our normal men can only cast two per battle, that's a lot of tiny fireballs.
Anti-Anti Magic Amulet: Create a variant of the Anti-Magic Amulet. One: a little bit less coverage, only enough to cover a single mage. Two: this amulet, instead of blocking casting, blocks the blocking of casting, at least by our own amulets. It's doubtful as to whether or not it'll work on their staffs, but at least it'll let our mages cast without fear of being cut off from magic by our own amulets, and it's a start, anyway. ((Closer to a new design warning.))
Improved Normal Arrows: We are farming trees for wood arrows, right? Why not breed the trees to make better arrow wood? Okay, generally, tree breeding takes a while, but we're mages, we'll make it work.
Bomb Arrowhead Molds: Using a mold for our Anti-Magic Arrowheads, we can more cheaply and efficiently manufacture them(by letting the spell to create the arrowheads fill the mold instead of constraining it manually).
~~~
Dragon's Fireball: 0
Study Staff Material: 0
Micro Fireball: 0
Anti-Anti Magic Amulet: 0
Improved Normal Arrows: 0
Bomb Arrowhead Molds: 0
I particularly like the molds and the Dragon's Fireball ideas.
~~~
Order : Do not deploy anti-magic-arrows. They are garbage.
Well, we do have anti-magic amulets for that... maybe the anti-magic amulets prevent the arrows from detecting magic? Hopefully?