As the guests arrive, spring on them which character they are going to be – give them the invitation information, and send them off into a room in your house where you’ve collected a series of props / costumes and costume accessories and let them find something appropriate to wear/use during the evening.
We often do this when we’re team building, as very rarely will a company group have time to get changed and turn up ready for a participative murder mystery in fancy dress, rather we look down the invitation list, work out what props might make a good extra for the characters listed, and then go shopping for costume accessories. We then give the group a suitcase of necklaces, hats, wigs, and other costume accessories, and ask them to find something appropriate to their role. E.g. the Reverend might create a white dog collar for himself out of paper and a paperclip.
Once everyone is there, you can just start the party in the normal way. The lack of preparation means that everyone is in the same boat, and no one feels nervous as they’re all just thrown into it.