Doors closed early and missed flights, anything I can do?

I know Frontier is not known for customer service, but today I really got screwed by them. Due to a number of cascading delays on public transit, I was cutting it REALLY close. I did not want to miss this flight so I was absolutely booking it through the Denver airport. I probably ran a 10 minute mile from the metro rail to my gate.

Anyway, my boarding pass stated doors close at 4:19. After my full speed mile run I got to the gate and the display showed 4:18, and then changed to 4:19 with no gate attendant in sight. I looked around frantically because I still have 60 seconds right? The gate attendant finally appears 20 seconds later, and said sorry, she's already closed the doors. Well I was there staring at already closed doors at 4:19, so you closed them early. How is that fair?

I went to their customer service desk, and the rep said sorry, you were 5 seconds late, please give us $190 to reschedule your flight. I had to pay up because I needed to get home. Any chance I can recoup some of this cost if I make a big enough stink?