Official advice is that you should carry your titre de séjour with you at all times but it is not a criminal offense not to have it (you cannot be punished for that, not even by a fine). The police does however have the right to keep you for up to 16 hours to ascertain who you are and whether you have the right to be in France (which you have, as you hold a valid long-stay visa, even if you are unable to present it). That would obviously be a major inconvenience and is the main risk in your situation.
Realistically, a check is not particularly likely but possible, and having copies of your passport and some receipt of your visa application should help. While the police could make your life unpleasant for a few hours if they really want to, the only purpose of this procedure is to give them time to obtain a removal decision against people staying illegally. So if you can convince them it's not your case or the préfecture can confirm they have a record of your visa, there is no point in keeping you any longer.
In the unlikely event that it comes that far, note that you have a number of rights while the police is holding you to check your status: You can ask to see a lawyer, a physician, an interpreter (if you do not speak French) and request that a friend or member of your family (only one person, though) and your consulate be informed of your situation (these rules are specific to the 16-hour vérification du droit au séjour but do not apply to the four-hour verification after a regular identity check, which is a distinct procedure, even though the latter can lead to the former). Additionally, you should not be restrained unless you behave dangerously or uncooperatively and you should not be mixed with people who are suspected of a crime (garde à vue). At the end of the investigation, the police officer in charge will write a report (procès-verbal), which you do not have to sign if you don't agree.