Would the service breaker or sub-panel breaker trip first if they have equal ratings?
Your question depends on very many variables.

1) Are the breakers magnetic trip or thermal?  Which is which?

2) If you are using thermal breakers, which breaker is exposed to the highest ambient temperature?

3) What is the current profile of your high-current loads?  Relatively constant or fluctuating?

In general, it is a crap shoot as to which breaker trips first when two identically-rated breakers are in series.  The breaker with the smallest amount of internal thermal heat-sinking will usually trip first but there are no guarantees.

Your best bet is to either change the breaker that you have access to down to 100 Amps or increase the feed breaker to 150 Amps.  I'd have to check, but I think that your 3/0 feeder cable is rated for at least 150 Amps.  Check with a qualified electrician to make sure.