The tricky thing here is that arcane ward is an academy card, and academy is not the same game as arena. Things work differently in academy, including how you cast spells, and this card was designed to function in that environment. In academy there isn't a pay costs step or a counter spell step. Casting spells consist of only two steps, declare and resolve. Under that framework, arcane ward was designed to trigger during the declare step, and without any official statement to the contrary, that's also when it will trigger in arena with how it's currently written.
Thus, arcane ward triggers during the declare spell step. The seeking dispel is discarded. You never spend any mana. And afterwards you may reveal bear strength whenever you'd like.