Won't Restart after quick trips on hot afternoons

Our T&C won't start when the temps are over 93 degrees outside.  It normally happens
after quick trips (5-20 minutes) when the car is then restarted within 30 min.  It has
however happened when the car has sat for over 10 hours in 95 degree heat and was
attempted to be turned on.   There is no noise when the key turns over, every error light
on the dash is on as well as mileage etc. but all gauges are dead, the radio, power
doors & windows all work.  It gives the Error  "Service Tire System Soon" and will give
you the direction & miles till service but not the outside temp or the running hours.  
Then if we keep trying to start it about 15-30 min after the first failure it restarts and
runs normally.  The drive train module has been replaced as we have had it at the
dealership most of this month. It is under warranty still but they can't figure it out and
say we have pay for any repairs or replacement parts they use that turn out not to be
the culprit. We live in Florida where high temps are the norm with two small kids that
cant wait in their car seats for this thing to decides to start and we can't afford a new
one right now. Any help is greatly appreciated.