The current is also zero twice a mains cycle. You could zero the integrator every half cycle. Sinceyour load is not inductive, the zero crossing of voltage on the secondary wouldwork as a trigger.

Anyway, in development, why not TRY it, and stop thinking too deeply - you may find that since the integrator runs with an AC source, the thing doesn't saturate, since it is integrated up one way then down the other.

Steve