Too be honest, I have no experience in either situation at this point. However, if it was me, I would make the appointment as soon as I could and state my case directly at the passport agency. I'm not a big fan of waiting to the last minute to see if they could help me...too stressful. If they tell me to come back until 2 weeks prior to my trip for it to be considered "urgent", then so be it.
In a matter of fact, I literally called them the other day asking how fast it would be to make a passport for her if I provide them all the information they needed. Their answer was "I can't tell you for sure, it depends". I then asked could I get an average or ballpark figure since I would think this happens to people all the time. Their answer again "I don't know". Bad customer service and no accountability as you would expect from our public agencies....it's just sad.
Hoping other people with experience can chime in and offer better advice.