How strict are US passport agencies with their 2 week / 4 week time limit for travel?

How strict are US passport agencies about the 4 week limit appointment limit?

I'm a US student traveling to Austria for study abroad in 5 weeks and need a new US passport as well as an Austrian visa D, and the Austrian consulate says it takes up to 3 weeks to process visa requests. I need to have my new US passport issued before I can apply for my visa, so in order to get my visa in time I need my passport as soon as possible.

However, the US passport agency scheduling system is very strict regarding when you can come for an appointment. You are not allowed to make an appointment for an expedited passport outside of 4 weeks of your date of travel (probably to prevent non-emergency cases from clogging their system.) I called the passport hotline and the woman literally wouldn't allow me to schedule an appointment outside of 4 weeks despite my pleading. I'm currently 5 weeks away from my date of travel, and waiting another week so that I fall inside the 4 week travel deadline really doesn't seem safe considering I have to wait up to 3 weeks to get my passport and another 3 weeks to get my visa. I don't want to press my luck here.

Even though the woman wouldn't schedule me, I was able to go online and get an appointment at a passport agency for this Tuesday (still 5 weeks out from my travel).

If I go on tuesday and explain to them that the austrian consulate takes longer than usual to issue visas (3 weeks seems pretty long given some of the stuff I've read) and that I need my passport expidited to account for that, will they make an exception and grant the passport even though my flight more than 4 weeks away?