hi,
Actually, it is better to wear long sleeves. The trick is to layer, not to get a thick jackets. It may be too cold for short sleeve in the shopping areas or shops, but too hot for jackets. My kids attire for winter is long sleeves t-shirt, followed by pull over, then follow by wind proof/water proof jacket. Of course depends on the temperature, they may just wear long sleeves and pull over during the day. And plus jacket at night. And in the shopping centres or heated areas, it could be just long sleeve shirts only.
For my daughter. Her winter attire is, plain long sleeve inside, outside can be a nice t-shirt or dress. If she wearing skirt or dress, it will be with leggings or tights. Anyway, if your kid is scare of cold, you can wear leggings inside, and pants on the outside.
If you want to make use of your singapore clothings, actually the easiest is just pack 2 or 3 long sleeve plain shirts. I not sure how much they selling in sg, but they are like 6 dollars here. Wear the long sleeve t-shirt, then wear short sleeve t-shirt, polo, shirt or dress over it. All the kids/adults do it here. If boy, just wear long pants. If girls, if wearing skirts or dresses, buy tights. But if your kid 3-7, it is just easier wearing pants or jeans. Or else you will get dirt patches on your leggings and tights. And it is really difficult to wash them off. And some young kids don't like to wear tights/leggings.