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If the fuel injectors are original they tend to fail when hot. The FI system is not sequential it is "batch fire" = if one injector shorts none of them will work = no start when hot. The injectors may be heat soaking during hot engine shut down.
If the engine has any misfire, stumble etc when hot it is an indication that the injectors are failing.
Ohm test each injector when hot you should have 14.5 to 16.5 ohms. If any are out of spec leave them disconnected & try starting the engine. If it runs (with a miss fire) the disconnected injector(s) are the issue.
I may have missed something, are there any lights on the DIC when it fails to turn over? If it was VATs related, IIRC you'll have a security light on.
As far as another key, if you only have 1 get another one cut and insure the resistor value matches the one you have (Dealer cut one for a friend and the wrong blank was used).
Now if you can duplicate the issue on demand, I'd get a remote starter first. Drive or whatever the car to set it up for it failing. snap on the remote starter, then first confirm it won't turn over using the key then try the remote starter. If it turns over that way you've removed part of the various systems from the equation (like a bad battery or not).