My 2c’s having been through a similar scenario:
a) Have assumed that you are using QF both ends which means you will be mainly around Terminal 1 (EK departs from T1 as well?). You could easily walk between all 3 terminals airside, just need to plan the timing (http://www.changiairport.com/etc/designs/cag/clientlibs/map/img/landing.png)
b) Plaza Premium lounge access using Priority Pass gives about 2hrs access per person, unofficially you may be able to stretch it to 3+hrs.
c) I have also used the T1 Plaza Premium Lounge, and noted that they did have a small “family/resting” room about the size of a medium office cubicle. Could not enter as it was locked, and staff suggested I pay first before having a squizz. Looks like SGD 180 for 3hrs, from the website: https://www.plaza-network.com/location_detail?airport=Singapore Changi Airport&city=Singapore
d) What w-hiew has suggested above may be worthwhile, if you are keen to bring the family in tow out to city, which requires clearing and reentering immigrations + waiting for the cabs etc.
e) Long story short, assuming the timing & arrangements are not convenient for all of the above; can I personally suggest using one of the transit hotels? (link below) That way I was able to get my family to sleep for a proper 8hrs (T1 Family SGD 160 + SGD 40 for addtl 2hrs) without the hassle of immigration and luggage, as well as a proper shower, rest & breakfast from a midnight arrival at SIN.
I would personally prefer to arrive at my holiday destination refreshed and healthy, rather than jetlagged with my immune systems worn down but horses for courses eh?
Hope this helps, have a great trip.