If I was wanting to do something like this, i'd be looking for a cheap room in downtown Singapore. Something like the Ibis Novena (which is v. close to Newton) springs to mind, and there's another Accor property on the airport side of town (the Grand Mercure Roxy) - pity as you've just missed their mega sale, which would probably make them cheaper than the transit hotel!
Also, I can't see the merits of going to both Lau Pau Sat and Newton, especially at that time of night. They're both open 24hrs but I always thought a lot of stalls closed up rather than going right through.
If you wanted a 2nd 24-hr experience then the Mustafa Centre is probably a good bet!
And you want to account for how you'd get from the City back to the airport at 3am (notwithstanding the possible issue with being able to check-in). The MRT will have stopped, you could I believe take a bus (1hr journey), or otherwise it's a taxi with a 50% surcharge for night hours tacked onto the bill.