I am a Mexican citizen with a valid US-issued border crossing visa (CBX). I do not have a Mexican passport but I do have valid ID to prove I am a Mexican citizen. I will have a valid boarding pass for a future flight from Tijuana to Veracruz, Mexico. Can I use the CBX for boarding from the US side, or do I need to go to Tijuana through the normal border crossing and then go to the airport?
CBX (assuming you are referring to the Cross Border Express ticket) is NOT a visa, it is simply a ticket that allows you to use the sky walk that connects the USA check in area to Tijuana Airport.
To cross the skywalk either direction (into USA or into Mexico) you need your passport, boarding pass, CBX pass and any necessary visa. The boarding pass needs to be for a flight departing within 24 hours or having arrived no more than 4 hours earlier.
Theoretically you should be able to check in on the USA side. Your flight is a domestic Mexico flight, so technically your Mexico ID should suffice for ID, since it will be Mexican authorities who do any security screening for the actual flight not TSA.
But then there is the issue of a passport being required to use the sky walk. You might trying emailing firstname.lastname@example.org for clarification on your specific case.
Might not hurt to look into to ironing out your passport issue in advance, getting it renewed or replaced, etc.