Mexico City used to process DCF, as did Monterrey (and maybe another one I'm forgetting), but it seems that changed during the added DCF restrictions in 2020.
Others have chimed in to say that DCF can't be done on a tourist visa, but I did just that in 2017. This, too, was before DCF became more restrictive so things might have changed now. I'll note that I had been living in Mexico for 7 years on tourist visas by doing visa runs (leaving and re-entering every 6 months) so I was able to provide evidence of my living there (lease, bank statements, passport entry stamps). Back then I was able to call the embassy and the person on the phone seemed to indicate that it didn't matter, and that they would consider my case as long as I could prove I had been in the country for 3 months. Again, things might be different now.
I'll also add that I did get apply for and get my mexican resident card just in case, bit I ended up not needing it as our case was approved before I received the card. The process to get residency was simple. I hired a lawyer and paid a total of around $600. Total process took about 6 weeks. If the embassy is unwilling to accept your case then you might consider just applying for Mexican residency, though that may not work with the timeline for your new job.