Of the big cities, Bangkok would get my vote as well. As an alternative, consider Chiang Mai. A number of years ago I used CM as a base of operations for several months, and it was surprisingly cost-effective for regional travel. What you pay out in a bit extra airfare is more than saved in lower living costs. A more manageable and less-expensive city than Bangkok (which relatively speaking, isn't particularly expensive itself) yet with good infrastructure. A good city for kids.
SIN would be my next choice but only if budget isn't an issue. Rents aren't cheap in Singapore, and most everything else is on the expensive side as well, by SE Asia standards. It's a good hub though.
KUL is possible, but for God's sake, don't let the fact that it's Air Asia's hub be the critical factor that crowds out all other considerations in your thinking. Don't pretzel-twist your plans to fit around the Air Asia company.
HKG is very high cost for rents, and not as good for purposes of travel focus on mainland SE Asia though OK for southern China access. It's a nice place to visit while in the area, but as a base of operations, definitely not when compared to the other alternatives. Skip this one.
BTW, neither India nor China is are in "SE Asia."
Nor do they lend themselves to 3-5 day excursions from a SE Asia base, once you figure in the cost of visas and travel. If either India or China is a priority, then I suggest you block out a specific time period (2, 3, 4 weeks, etc.). And then either front-end or back-end that time, so your base hub rental does not overlap--you don't want to be paying double for a place you aren't at.
I have similar doubts about 3-5 days for Indonesia excursions. In short, while I think your base hub strategy could work as a concept,
you have way too much geography to deal with in "3 months or so." You'd really need more like 6-8 months for all this and it's really not possible to do the entire thing as short excursions. I suggest that if SE Asia is your focus, that you dispense with India and China and well, FOCUS. Or, reconceptualize this trip and just make it a continuous journey rather than trying to settle down at a base. You could do a hybrid and plan a sequence of mini-hubs of two weeks per hub, if that would help settle the kids slightly.
Visa strategy is an integral part of planning, so get familiar with the requirements of the different countries and what might be available to you.