The hardness of the steel H13 may be varied from 46 to 56 HRC through a variation of the post-quenching (tempering) temperature. Of coarse, the fracture toughness decreases with hardness increase. You can start you brief applicative research from the middle of this hardness range, namely at 51-53 HRC (tempering at 575 cent deg = 1065 F) and try to exploit. If any adjustment will be required - do it. Take into account the operative temperatures, stresses and friction of your forging process.
Regarding nitriding: it indeed improves the wear resistant at the operating temperatures (and, therefore, the lifetime) of dies made of this steel, if you adjust the lubrication.