There doesn't seem to be any option for a new passport for for travel in 4 weeks. There's expedited which takes 5-7 weeks and urgent for flights within 14 days. Not sure what the best path is for a wedding on April 30.

Call them on the phone.... Speak to an agent and let them know that your reason for emergency travel is a wedding... If they say that works.... Set an appointment with them within minimum of two weeks of your before your travel and maximum of 5 business days before your travel time.. You need an itinery with a ticket confirmation number before they can set an appointment for you..

Apply and pay the expedited fees now ASAP and the Call in two weeks prior to travel for an in person appointment if you haven’t received it on time. They only book for those traveling with in two weeks but can get you a passport same day. They say 5/7 but I have see. The usual trend is 2 weeks if no problems and if it goes down to an in person need, they can pull up your already sent in application to finish it.