Things like this is the official and (I'm pretty sure) legitimate reason that
Game Copy World exists. Search for Bioshock and download the no-dvd fixed exe (I checked, they have that one). Backup the original game executable, then replace it with what you downloaded to eliminate the check that's failing for you.
Yes, this could be used to make a pirated copy work. You would have to obtain said pirated copy from somewhere else, however, and GameCopyWorld does have legitimate uses such as this one, along with an emphatic legal notice and anti-piracy warning that takes up almost the entire entry page.