Terrible idea imo, build first then brood them in the coop. My chicks are fully feathered and off heat usually in 3.5 to 4 weeks brooding in coop with overnight temps in the mid 30's
People should not put the chick before the coop
I would run out the lease, let the rental company pick up the "coop" and hens in the fall. Meanwhile start your own build, go big! Make it a walk in, or buy a shed you can modify then get a proper flock of 6ish chicks in the spring, get the chickens you want....