Dodge Stratus 2004 sedan has trouble starting under very specific circumstances

Sounds like lack of fuel pressure at the rail. has this been checked? also was the fuel filter changed?

The idea that the cat only fails to start at a certain location sounds coincidental but until you know, we don't know. If it is hot outside in late afternoon that might be heating some loose starter connection causing it to fail. You might try to jump the starter with a screwdriver the next time you see the problem. This will bypass the ignition switch and might tell you if the switch is bad. This is a real mystery, please let us know what you find out.

The mechanic said that he does not want to check the fuel filter or pressure yet due to the fact that the car is acting perfectly normal and starting just fine. The problem has been going on for a year now, it started last summer during the high heat. I assumed it was an issue with the hot weather, but the problem continued through the harsh winter and continues to occur since then, but only under those very specific circumstances. The mechanic is currently going to keep the car in the bay overnight and continue to let it sit - they had it sit outside in the heat, inside in the bay, up on the rack, but the car is still very happy and won't replicate the problem for them. Everything that they have tested so far works great. Once I get the car back, I am going to park it in a different parking lot at work to see if the problem still occurs at the problem time (4 - 4:30 PM) in the new location. When I am at home, I am going to start it at 4:30 PM to see if the problem occurs at home (which it never has). I am very baffled by this, after an entire year of this very consistent odd problem, and after the car suddenly gave me so much trouble starting after work yesterday. Thanks for everyone's answers so far, I will keep you posted.

Have air filter checked if it is clogged vehicle is starving for air

Sounds like a vapor lock to me