Sentosa is q. far away from the main activities on the Singapore island. There are some activities and theme parks on the island itself but to travel to city, you would need to take the bus or train,which is about 20-30 minutes away by train because you still need to change train. There is however a big shopping mall that is newly built and just across the Sentosa island called Harbour front. Its a bb friendly place with some international boutiques and a whole floor of kids stuff.
Usually Sentosa is a place for getaway from the busy city. So depending on what you want from your holiday. A seaside resort getaway, i.e. Sentosa, or citylife, i.e. Orchard Road.
Along Orchard Road, there are international hotels like Shang, Ritz (expensive), Westin Stamford, Conrad. Most of the hotels should be bb friendly.
The thing about Singapore is that it is more bb friendly than HK and most places, even food courts have bb seats. So it shouldnt be a problem for your young ones.
Good luck and have fun in Singapore!
PS: heard it has been raining everyday in Singapore now so bring your brolly along!