Concerning the weight of Rock Doves;
John K. Terres in The Audubon Society Encyclopedia of North American Birds
(Alfred A. Knopf, New York, 1987), quotes two sources for Rock Dove weights.
One (Hughes 1970) gives 286 grams or about 10 oz. The other (Palmer 1949) up
to about1 lb.
I'd say they are about equal. Which answers the second part of your question
on which common bird weighs about as much as your cockatoos.

Martin Muller, Seattle
martinmuller at msn.com