"3 Hours" is a very common recommendation for international flights at many airports around the world - and it's almost always far more time that you need. Airlines like to get passengers to the airport early, not only because it gives more buffer if there are any problems, but also as it spreads out the "rush" period before the flight - you can imagine what the check-in counter would be like if the entire plane turned up exactly 90 minutes before the flight!
As long as you are at the airport 2 hours before you will have plenty of time (even 60 minutes is enough at some airports!). Given you're traveling with a young child and in an area you're not familiar with I wouldn't recommend cutting it any more fine than that. That's not to say that planning to catch the 5:52am train isn't a good idea - obviously the earlier you get there the less chance of something going wrong.
As far as easing your worries, the best recommendation I can make is to make sure that you've got multiple alarms set for the early start. An alarm on your phone, one on the in-room clock, and a wake-up call from the front desk wouldn't be overkill in a situation like this, and means that if one does fail for some reason (phone hangs, power outage in the room, etc) you'll still have something to wake you up!
Check with the front-desk in advance for any hints on getting to the train station - perhaps the entrance nearest the hotel isn't open until 7am, or before 6am you can't buy tickets from the ticket booth, etc.
But most of all, don't panic! If you plan on the 5:52am train as you are, then you've got some buffer in case of problems. Short of something major happening, you'll make it easily! And you can always sleep on the plane :)