Upvote:4
Your information is correct. The UK emergency passport is not an e-passport, so it does not allow you to use the US visa waiver program.
You can consider a couple of options.
1) Travel to the UK and make an expedited passport application.
2) Try to get an expedited visa appointment. One way to do that is to have been denied ESTA authorization. You may be able to get an ESTA denial by applying for it with your non-qualifying emergency passport, but I am not sure whether the application actually asks about that.
Another option, of course, would be to travel to Mexico without passing through the US.