It is difficult but so what?
If you want housing agents falling over themselves to offer you accomodation for your large family, then be prepared to pay for the convenience of not hunting down a place that matches your expectations and budget.
$3200 will still get you a decent house (maybe EVEN a condo) within 10km of the airport. You just have to make the effort to look for it. If you expect nice properties to be presented to you by prancing girls on bended knee at prices within your allowance, dream on.
Many MANY pilots working for other airlines based in SIN are happy to live in HDB accomodation and save the difference between allowance and actual expenditure. Is it so disagreeable to live in HDB?
Or is it a case of the sheep not wanting to mix with the goats?