We are in Texas (8b) so our experience may differ from yours, but we don't fertilize trees in their first year. Trees are settling in, getting feeder roots established, and generally adjusting to their new site. Fertilizer usually stimulates top growth, which is a stress on the tree under these conditions.
Watering deeply once a week is a very good idea, and ten minutes a day by hose doesn't seem like deep watering, but the guys at the tree farm ought to know. Bottlebrushes like moist soil, so a deep watering followed by enough watering to keep the soil moist but not soggy should make your beautiful tree happy.