Cutsomize a weapon for a pvp charr.

Is it possible to customize a sword i have on my pve charr for my pvp char anyway ?
i dont want do redo the pvp char becuse i have some good runes on him : /

its already customized.. if it wasnt you could just pop it in storage and use it on non-pvp characters

yeah.. but i want a sword i got whit my pve char by looting it, And i want to customize it for my pvp char so i get 20% bonus to damage.

If i customize it for my pve char i can not have it on my pvp char.