You have written a lot, but actually said very little. You have not provided any info about the project itself and are losing yourself in technobabble. If it works on one machine, then soemthing in the project itself is inherently fouling up, which could be anything from a specific effect to a certain audio/ video CoDec. Start by cchecking these things in the AE prefs, the system settings and the relevant section of your device manager. Also rather than vaguely telling us about fancy SSDs and graphics hardware it might be more useful to actual list the full specs of the computers. Even a trivial difference in processors or audio chips on the motehrboard can cause some of these wieird behaviors.
I'm no expert in Hardware whatsoever...thereby giving any info that I knew of in that area I was not trying to be fancy in anyway.
I will go back and check any other preferences that I can think of might effect the project on my machine..thank you for your response.
Did you ever solve your performance issue? Please let us know if you still need help.