Should I (the prospective second wife) contact the first wife myself without the man's knowledge?

Usually not advisable — informing the first wife is the husband's responsibility. Contacting the first wife suddenly can shock her and make matters worse, especially if she does not yet know. But if you suspect the man is lying about his marital status or his wife's agreement, verifying it is your right before making a lifelong decision. A better way: ask him to inform his wife first and show proof of the court application. If he refuses, that is a clear enough answer. See relations between co-wives.