as I said, I think it's possible to do the missions while on duty if you get into a different police car than your own.
Edit: argh, apparently that was a rare occurance, can't do it right now when testing anyway.
Is there a way to change the police computer-key for the mod yourself perhaps?
Edit again: Yay, changing the "action"-key ingame from E to another key seems to do the trick, E brings up the LCPDFR-computer while the new action key brings up the old computer.
Edit once again: Or i guess you could even add a new optional key for "action", so you don't have to get used to using a new key all the time, just press the optional one when you need the old police computer.
Edit once again, again: Having another optional key for "action" works fine, now I can use "E" for the LCPDFR-computer and various action and the alternative key just to bring the old computer up, vigilante missions seem to be working fine aswell, arresting suspects even works instead of killing them it seems!