If it is just a chipped OEM ECU, then chances are it may still have the flood clear function enabled. In that case, I would be tempted to give the flood clear a try. If the flood clear is successful in starting the engine, then the problem is likely over-fuelling during start ups. That may be a tuning problem or some other problem like excessively high fuel pressure during a hot start.
Since the ECU is essentially OEM, also try the service check connector to see if there are any stored error codes that might indicate what the problem is.
You mentioned 'some of the mods the car has'. What other 'mods' does the car have? You haven't switched to a different fuel injector by any chance? When cold, engines are moderately tolerant of a little more than necessary fuel delivery on start up; but, a little extra fuel can make hot starts a pain. Those other mods might be contributing to a hot start issue so those details would be useful.
My understanding is that the chipped OEM ECUs from Japan primarily modified the ignition timing map to presumably take advantage of the 100 RON premium (JIS standard) fuel in Japan. That would be about 94-95 by the R+M/2 method used in North America. According to Google, Oz also uses the RON rating method and the common premium is minimum 95 with up to 98 available. That is significantly below the JIS premium 100 RON; but, 97 RON is in the range of the typical 91-92 RM/2 in North America. So, if you are burning 97-98 RON there is no advantage to using the revised ignition maps since you are knock limited by the fuel. There is a down side to running knock limited with a revised ignition map. If the map is set to use 100 RON it is possible for you to initiate detonation with 98 RON fuel using the 100 RON map. Presumably, the knock sensors will detect the detonation and retard the timing. However, detonation will have to occur for the knock sensor to work. If you run 98 RON with an ignition map meant for 98 RON you are timing limited and you will never incur detonation if everything else is fine. The knock sensor is just there as a back-up safety system for a bad batch of fuel or the time you had to fill up with regular because they ran out of premium. You have moved the knock sensor from a back-up safety function to your first line of defence. Not a good idea, Particularly if your knock sensors are getting deaf with old age.
Personally, unless the compression ratio has been increased, I am undersold on the benefits of bumping the ignition timing to take advantage of fuels with higher octane ratings. Higher octane fuel actually burns slower than lower octane fuel so it needs more timing advance to complete the burn at the optimal point after TDC. You can tweak the timing to fix the burn time problem; but, if you have not bumped the compression ratio the big up-side is not there.
Rather than change the ECU, you might want to check out the work that
@MotorMouth93 and Honcho were doing with the characterization of the pre OBDII ECU. With a small amount of soldering work on the ECU and the correct software interface you will have access to the fuel and ignition maps. One of them might be able to supply you with the ignition map for a North American na1 which are set up to burn 91 R+M/2 which should work just fine with 98-98 RON in Oz. The big up side is that the start up and warm up configuration in the OEM ECU will be miles ahead of anything that you will get for a stand alone ECU because you do not have enough time / money to pay a tuner to spend the time to get that stuff sorted as well as Honda has it sorted (been there, done that, got the Tee shirt). Plus, the OEM ECU has error code reporting (although not as extensive as on the OBDII ECUs.