I live in KL but I'm familiar with the schools in JB.
There isn't much in JB. Singapore is a short distance but Immigration gets busy at weekends and rush hour, and it could take 90 mins to get to central SG.
Unless the package is very good, living in Singapore will be tight. Your package would likely assume you'd be living in JB which is much much cheaper than SG. You could probably get a house with garden in Horizon hills (nice area in JB) for MYR5000.
If you have a Malaysian Work Permit, you might not even be able to live in Singapore. (I'm not sure on that, so check)
I have a friend who lives in JB whose dh leaves the house at 6.15 and is at his desk off Orchard road in Singapore by 7.30. If he leaves later it can take hours. Coming home is a PITA
If you were working in JB and living in Singapore though you'd be doing it the other way around.
She moved from Singapore as their package was only enough to pay for a two bedroom shoe box in Singapore and moved to a four bed house with a pool in JB. Their kids are in Marlborough.
There are several new start schools in the area- Excelsior, Sunway etc. I heard there was a problem with the SLT at Excelsior and a lot of the initial staff left (not sure if true though)
Be careful of Fairview - I'm not sure about the JB campus, but the KL campus is notorious for treating their staff very badly. (same owners)
Another thing to bear in mind, is that International schools are mushrooming all over Malaysia. They are not aiming at the expat market though, which is actually in rapid decline, but at the middle class local market as the state school is simply not fit for purpose (60+ per class, institutionalised racism, badly paid demoralised staff, falling standards, rote learning and lack of resources)
A new start school would likely be 99-100% local students (apart from the odd staff child). This in not necessarily a bad thing, ( I teach in a lovely small school with 100% local students myself). But if your own kids are going to attend the same school, you might want to be aware they are likely to be in an extreme minority.