The problem is actually pretty simple. As soon as the DOS device driver (BTRDRVR.SYS) load, it finds the "helper" file (BTRVDD.DLL)in the C:\WINDOWS\SYSTEM32 folder as it is supposed to do. The Helper DLL then looks for the start of the Pervasive Client -- W3BTRV7.DLL, which then fails. As Mirtheil noted, the DLL is placed into the proper Windows path on Windows 8, but for some reason, Windows 8 cannot find it.
As a REALLY nasty workaround, you can copy ALL files from the Pervasive install folder (C:\Program Files\Pervasive Software\PSQL\BIN) over to the C:\Windows\System32 folder, and you'll find that your DOS app should now work.
There is likely to be a smaller subset of files that you can copy over, but you'd need to find these one file at a time using Process Monitor, like I did.