OK here's another issue. First, I wasn't hitting Apply on the new NAT settings, didn't realize you needed to Apply after hitting OK on the dialog.
But that didn't fix it. In fact, it made it so my server (in the VM) is no longer responsive. I think that was because the files for the program reside on the Windows machine and they're shared over to the VM via SMB -- didn't want the files to be lost if the VM got corrupted or something. After applying the NAT settings I can't get to the files that are on the SMB. Something is wacked here.
So I'm going to attempt Bridged. Later.