Call around to siding companies, warehouses, look on Craigslist free site. Someone will have old/free pallets. Pull out and reuse the nails in the pallets. You can build almost anything with pallets if you have enough time and patience and the right tools. I think you should "extend" the hen...
Go for it with the ordinance changing.
In the meantime, consider obtaining free pallets and constructing a private/fenced run. If they can't be seen, they can't be complained about. Put bird netting over the top, or a fencing panel to prevent them leaving or hawk attacks.