No crank no start

What did you do under the hood at the fuse box in order to get it to finally crank?

See if any of those fuse blew then.

Is the security light still on?
Does the fuel pump run when you first turn the key to run, before you turn the key to start?
Do you have a spark while cranking?
Why the engine start and run for a few seconds after a few shots of starting fluid into the throttle opening?

As far as I can tell you are just jumping the starter relay in order to get the starter to crank. That doesn't mean the engine will start, just crank.
I remember reading you replaced the BCM and also you said the radio was locked and also the security light is on. That make me believe the security system is preventing the engine from starting and most likely disabling the fuel pump and ignition.
Did you ever reprogram the bcm after replacing it so it learns or has the password from the ignition key? If not, I believe that might be all or at least most of your problem, cause I know you still have a problem with the starting circuit.
Actually it appears the BCM is not in that loop, anyhow try this to get that security light out.
With a master passkey III key (Black), turn ON the ignition, with the engine OFF.
Observe the Security telltale, after approximately ten minutes the telltale will turn off.
Turn OFF the ignition, and wait five seconds.
Repeat steps 1-3 two more times for a total of three cycles or thirty minutes.
IMPORTANT: The vehicle learns the key transponder information and/or passwords on the ignition switch transition from OFF to CRANK. You must turn the ignition OFF before attempting to start the vehicle.
With a master passkey III key, start the vehicle. The vehicle has now learned the key transponder information and the PCM has now learned the fuel continue password.
With a scan tool, clear any DTCs.

Also this is what I read about the radio being locked,
"A module which communicates on the class 2 serial data circuit which supplies VIN information to the radio is replaced and not properly setup with the correct VIN information for that vehicle
The only way to unlock the radio is by using a scan tool. "

I have no idea. I can give you the wiring diagram. Perhaps if you posted the instructions you are referring to that is telling you to install a resistor I could look at it.

I have read posts about inserting a resistor, too but I have not saved any links.

First of all, stop jumping the starter relay and figure out why the starter will not crank over with the key and fix it.
Then stop trying to bypass the security system and fix that. If the security light remains lit, then there is a problem with the security.
Begin by feeling if the starter relay clicks with the key on start, if not use the wiring diagram and test if the fuse #27 (IGN 1 Main) has voltage with the key on run and start, if not trace back and find out why not, cause it goes through the ignition switch. Before that the IGN 3 (#19) fuse. Then you also need the "In Key" fuse to be hot, it feeds the ignition switch, too and when the ignition switch is on start it feeds the PCM which actually is what activates that starter relay.
Then when you get the key to work the starter, maybe just maybe you would have also fixed the security problem, if not fix that. Try a spare key, try a password relearn but if turning the key to run after ten minutes the security light never goes out, you will need to read the trouble codes and see what refer to the theft or security system then work on that. It is not easy to bypass a theft system or people would be stealing cars left and right.