Search the homes and/or workplaces of the 4 suspects, as well as the crime scene. There is no court system to grant search warrants, so you'll have to do a little sneaking or break and entry to poke around. Once you have something more substantial than rumour and speculation (see spoiler for a hint), the relevant suspect will be more forthcoming. If you had Pet Pal, that would open up a possible solution where Pickpocket could come in handy.
