Upvote:4
Update (as an answer because I don't have enough reputation for comments)
I ended up locating my green card a few hours before boarding so can't give a definitive answer to this question!
The airline was extremely helpful. I spoke to a customer service rep from the airline who felt that with a police report and copy of the card I might be allowed to board, but they were in the process of calling superiors when I got the card back so wasn't able to put that to the test. An exception would definitely be against official airline policy.
The US consulate was extremely unhelpful. They repeatedly refused over the phone and email to give me any information about boarding foil turnaround time, expedited processing or other immigration options without first paying the US$300 boarding foil fee, being contacted a few days later with an appointment slot some time in the future, and flying to the consulate in person.
I highly recommend not losing your green card while overseas in the month before returning to the US.
Thanks for the comments everyone. Sounds like entering as a visitor is bad idea, but you might be admitted as a LPR if you can get to the border.
(My country isn't on the travel ban list, I was away for < 6 months, my flight was ~12 hours after realising the card was missing.)