Do you have a spare bathroom? I did, and lined the bathtub with plastic, then newspaper, then 3" of pine shavings. I moved the tension-rod from the shower curtain to the center of the tub and hung their light from that, on a chain. I kept that bathroom door closed, and occasionally would run the ceiling fan to clear it out.
When I moved them outside, I un-did the tape around the edge of the plastic, lifted the whole thing up and out by the edges of the plastic, and dumped it in the garden. It worked really well, and kept the odor (which was minimal) to one room of the house. I did have to tape one sheet of plastic up about waist high as they got older, to keep them from jumping out of the tub. This hung inside the bathtub a few inches so they couldn't sneak out or perch on the edge of the tub. I either untaped the plastic barrier or reached underneath it to tend to the chicks, food, water, etc.
Some food-grade DE or "stall-dry" in the shavings would help even more. Their turds dry up to nearly indistinguishable.
Good luck!