Update to anyone who cares...I wrote a polite but insistent appeals letter (in Arizona there is a 3-tiered appeals process) arguing that these should be declared emergency services, and thus subject only to the associated $150 copay and no deductible (this was not clearly outlined in the materials I had from my employer, but I discovered it in my research).
This was in December, and I just now got a bill back from the hospital saying "XXXX paid by insurance, $150 remaining balance due" for the second event (the simpler one) so I'm hopeful that I might be able to get out of this mess for $300!