The reason there's a night 0 is so that people can get information so that the first day isn't just everyone going no lynch because we don't know anything. While we may not have any information at the moment, we don't gain anything from no lynch. We learn absolutely nothing at all, while the mafia get another night to go kill someone and do what else they can do. Especially since we have a cop who has already roleclaimed, I don't think it would be a good idea at all to no lynch. What will you do tomorrow if no one has found anything? Just wait another day/night phase?
At this stage we outnumber the mafia considerable so a random lynch is more likely to harm us than help us. If we have nothing to go on then we shouldn't bother.