Well if grass won't grow I can't think what will flourish. How big is the area? Its best to leave an area of unplanted soil round trees so perhaps you might consider using gravel or pebbles? You couldn't use a membrane underneath if yoU want to grow bulbs though, so pebbles would probably be easier to keep weed free. You could always stand containers on the pebbles for summer colour.

Don't use gravel - cats often treat that like cat litter, but larger stones are fine, especially pebbles. Cats only use soil if its been dug over recently and is nice and loose - once it settles back down again, they won't bother to go there, so in the meantime, if you don't want pebbles, lay prickly branches over the soil.