Turns out this nesting in vents thing is a common problem. My favorite of the proposed solutions comes from Calvin W. Schwabe:
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STARLINGS IN CRUST (Etourneaux en croûte à l’ardennaise) / FRANCE
Remove the backbones from some prepared starlings. Rub them with a mixture of salt, white pepper, and mixed spices. Stuff with a bread stuffing containing the birds’ livers, some mashed juniper berries, and, if available, some liver pâté and truffles. Wrap each bird in a piece of pig’s omentum. Pack tightly in a shallow baking dish on a bed of the backbones, chopped onions, and chopped carrots, all browned in butter. Paint the birds with a lot of melted butter and braise in a hot oven for about 10 minutes. Unwrap the birds and place them in a large bread croustade that has been buttered, “melba-ed” in the oven, and sealed with a paste made by blending in an electric blender some fried chicken livers, mushrooms, and egg yolks. Bake in a moderate oven for a few minutes and at the last minute pour in a sauce made by reducing a cup of sherry added to the braising pan, straining, and adding a cup of demiglace or other rich brown sauce. Garnish with some pieces of truffles lightly sauteed in butter.
STARLING STEW WITH OLIVES (Karatavuk yahnisi) / TURKEY
Fry some chopped turnips and carrots. Add a little stock and a glass of red wine. Place some starlings or other small birds in the pan. Add a thin purée of boiled potatoes mashed with beaten egg, dry mustard, and some stock and a little beer. Cover with stock and cook for about 30 minutes, adding some ripe olives near the end.
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…in the end, though, we decided to just ask the landlord to nail some wire mesh over the vent opening. Maybe next time.