The water doesn't even need to be warm. I've re-used the same oxyclean/water for several batches of bottles to delabel them (read: Sitting at room temperature for months) and the cold water works just as well.
The bottles being filled will actually be convenient because they will sink but still stay upright.
I can't speak to the bottle caps. Maybe sprinkle the oxyclean into the bottom of a rubbermaid storage tote, arrange the bottles so they stay upright, and fill with water until the neck-labels are submerged, but the caps aren't??
*Edit: Discnjh beat me to it