There is a glitch for this. You can start on Stage 4 (Factory) and complete the game from here to unlock this trophy. For that you should save as many workers and cats as you can. Time and health are very important for this run. You must not purchase anything from the store. You should also unlock
I have read that this glitch may not work if you have purchased a certain stat upgrade (unsure which one), so I did it without any upgrades.
I have also finished this on the PS4 from the beginning of the game. If you want to do it from the start then below is the guide for a proper run:
With enough practise you should be able to get through every level with enough time and health that you don't need the powerups. Buying powerups means anything in the shop. You are allowed to use the search function when it pops up to loot whatever is in the map.
I checked the shop on each run as I worked towards the last harder trophies. If it doesn't have anything useful in the first shop then I would try for this. You should try to save everyone you can to gain as much extra time as possible. Only do the missions that are not too time consuming.
With some practise this becomes fairly easy. Just make sure you are using the hose before entering a room to clear out the closest fire, then going in with the extinguisher and finishing with the hose.
Took me just under 20 hours of play time to get good enough that I could clear everything out without running out of time.