you most likely have a bad CAS, that electr sender on the driven at end of camshaft. (left cam rear? IIRC)
and are infamous for heat soak fails.
yours starts and runs , cold every time, right.?
so you have 2 paths.
1: prove its bad ( tests)
2: swap it and pray it is.
like i tried to say post 2. (your situation and symptoms)
1: failed starts can be caused by many reason.
a: loss of spark ( CAS above)
b: loss of fuel.
c: flooding.
did you try to start he car hot; when dead, by using WOT? (wide open throttle cuts fuel , unflood mode of ECU)
did you check spark when it dont start? (that be the very first check)
did you try test fuel, if spark is good (failed start) and then tried test fuel? (if sure not flooded)
tests: this is like 3 tests. i could list more but those are the prime fail paths.
truly no body has to guess. ? why guess?
a scan tool will reveal the ECT and thermostat issues pronto.
it is your money and you can spend it, guessing or testing .
time or money , I sure cant say what is best only you can.
$73 plus ship is new CAS. BTW
cheers