Wait , game is out from one week , and..

[Katagai]

It costs tier 1 allagan tomestones (one blue stripe), which do not have the 300-per-week limit. You're able to get these tomestones for exchanging gear with cracked materia slotted in it to the calamity salvager.

My HQ Hart Guisarme with 3-4 materia in it gave me 500 tomestones, for example. By exchanging my other gear with 1-2 materia for roughly 50-100 tomestones each, I was able to afford the Darklight Cuirass (800 tomestones I think).