I browsed forums and found this solution, it works for me with Origin! And it appears that root cause is long name for installation, which Origin apparently cannot handle.
1) Browse to:
64bit OS: C:\Program Files (x86)\Electronic Arts\Battlefield Bad Company 2 Digital Deluxe Edition
32bit OS: C:\Program Files\Electronic Arts\Battlefield Bad Company 2 Digital Deluxe Edition
2) Select all -> right click -> Cut.
3) Go to the root of your drive (e.g. C:\) right click -> New Folder (I called mine BFBC2)
4) Go into the folder you created, right click -> Paste.
5) Browse to __Installer\DISK1 (in my case C:\BFBC2\__Installer\DISK1)
6) Double click EASetup.exe select language and install as normal, make sure to install Punkbuster when given the option if you plan to play online.
7) Go back two directories, to the main directory (in my case C:\BFBC2) select all -> right click -> cut
Browse to:
64bit OS: C:\Program Files (x86)\Electronic Arts\Battlefield Bad Company 2
32bit OS: C:\Program Files\Electronic Arts\Battlefield Bad Company 2
9) Right click -> paste.
10) Close EADM and then reopen it, it should now say "Ready to Play" next to the game, click the button and enjoy!