Neither MPs or regular Army troops are AFAIK trained in the application and details of civillian law; the rights of the individual prior, during and after arrest, for example.
The problem with 'removing paperwork' by using MPs (for example) is that you are also risking removing accountability or oversight. That's one of the reasons, i believe, the military isn't used for these tasks. If they have a shortage of people, they should recruit more policemen. If it's down to too much paperwork, fix that problem. But they can't claim it's a good idea to compund the mess with 2 disparate groups of law enforcement, trained in very different ways.
(for example - who has the final say in policing in this type of case - the constable or MoD?)