Pamela, you might be able to just pull up the weed netting, with the weeds and all. Shake it off to leave the mulch there. A thicker layer of mulch at this time of year might help your weed problem stay under control.
Yes, I get lots of sunflowers and millet plants under my bird feeders, too. Weeding is just a fact of life in gardening. If it wasn't the bird seed some other weeds would have come up. Birds bring seeds and poop them out onto the ground as well. Fertilized weeds.