After playing with iTunes a bit, I found a pretty simple way to do it. First I clicked the triangle to expand the podcast series and then deleted the missing episodes (the ones with the exclmation) by selecting them and hitting the backspace key. After that, I collapsed the podcast series by clicking the triangle, and then expanded it again while holding the option key (I think it would be shift on Windows), causing iTunes to update the podcast and add back in the episodes I just deleted.
The re-added episodes had the 'Get' buttons again, allowing them to be re-downloaded.