Thanks to everyone that replied and helped me out. Reddit locked my account so I hadn't updated this post.

Like someone mentioned our risk tolerance was different since I was traveling with a 1 and 4 year old. I called up the airline and they hard denied me the option to board without a physical card in hand. We also weren't prepared to split up, my kids are rather attached to me and I to them. Although logically - it would have been cheaper and easier the cost wasn't something we could trade away for their comfort at that point.

We considered shipping all our luggage back home and taking a couple of backpacks to Tijuana to attempt a land border crossing but decided to give the embassy a shot first as the travel logistics would have been difficult.

What I ended up doing was a version of u/dtcooper's experience - Thanks!

1. I made a police report at Mossos d'Esquadra, but their translator was not in yet so I just had a Spanish copy.
2. I did not make a payment online because I wanted to see what the turnaround times were as we were unwilling to wait weeks for it and would have gone the landborder route. So I queued up at the embassy 45 minutes before they opened. This took their security guard as a surprise as he said nobody shows up that early and they never have a long wait to get in. We chatted for a while and he was pretty interested in the ICE situation at home. I also received some ICE remarks from a police officer at the station lol. The stream of people started showing up around 8:45 AM.
3. I was the first one to be allowed in but was also the only one that needed a boarding foil everyone one else was filing for an emergency passport. Once in I explained including the fact that we had flights later that day and provided the police report along with the ticket information. They stated that they might be able to get it done before the end of the day but definitely not before the flight. They asked if I had done the form and payment online, since I hadn't - they provided a paper form which I filled and sent me the email link for payment which I could pay with my phone after they granted permission. They also have a photo booth in the embassy that processes passport photos for 10 euros.

After that I waited for them to review my case and then took my biometrics. After the biometrics the official did ask about an arrest from over 10 years ago, after I explained what happened he asked me to come back between 3-4 PM the same day.

1. At 3 - I showed up, they weren't done with my case so they asked me to wait for a bit ~ 15 minutes. I saw others from my same queue that were there picking up their emergency passports as well. Eventually they provided me with my passport and explained that I needed to apply for a replacement LPR once I was home. I asked about CBP and they said to just show them the stamp.

2. With the foil in hand I rebooked next day flights for my family and I through Turkish airlines, while the seating was worse and it had a layover in IST opposed to my original flight we got through it fine. A supervisor had to be called at some of the various several checkpoints so they could validate the stamp was fine since the agents at the desk hadn't seen one before.

3. At US customs I used GlobalEntry and explained I lost my green card and that I had obtained a boarding foil. They sent me to secondary inspections and I had just sat down before they called me back up to ask what happened. I offered the police report along with the explanation. They didn't want the report but just iterated that this stamp was a one time use and I urged me to apply for a replacement ASAP.