yes the bose is a free air sub if you cna call it a sub , as youve popped it out im sur your now well aware why it sounds feable !
a 10" free air would drop right in with an mdf baffle to mount to , only issue is finding one in your price range as they arent very common so you may end up going over budget or very cheap.
just what ever you do dont be tempted to try a non free air designed sub as youl wreck it with no enclosure to provide the suspension.