How about a box?
At our vet, they use a big, clear plastic jug to weigh the birds. Lots of holes it in, of course. It's actually the jug that the wrapped peppermint patties in the lobby come in, so the opening is really wide.
Not to say that you then won't be chasing around a box...
Tare the weight of the box before you put the bird in, of course, or if you like the mental exercise, you can subtract the box weight afterward.
What kind of scale does everyone use?