you could move it to a box somewhere, not near lights preferably, I used a file cabinet once
you could clear out a another closet for a few weeks, maybe kitchen cabinet, pantry, shower

as for tent material, how big are you thinking. it might be cheaper to find the smallest grow tent you can, as I've found that DIY with these materials actually isn't cheap.

a cheap way maybe a box, paint the outside reflective of foil, since you like foil, line the inside with duct tape to fill the cracks.
but if your light is strong I wouldn't put anything under it that would spark (cardboard)

maybe a ice box, like a party cooler, ice chest (for camping),its square. stick it on its side, (hey you have a little door too), and you could put small plants ontop