A couple of suggestions for you.
1) a section of the Munda Biddi Trail in Dwellingup (a bit more than an hour from Perth). Specifically from Nanga to the Bidjar Ngoulin camp. It is approx 12 - 13 km's from the Nanga camp ground to Bidjar Ngoulin. The terrain is relatively flat with a few very short and sharp inclines/declines. The track itself is hardpack summer or winter, though in winter the centre of the track can get very slippery. You could start in Dwellingup and ride all the way out to Bidjar Ngoulin but there are some serious hills between Dwellingup and Nanga.
2) Base Camp at Nanga and simply explore the trails, which include the Munda Biddi and Waterous Loop.
3) Camp at the Marrinup Townsite camp ground and ride around the Marrinup cross country (XC) track, I reckon your kids would really enjoy that track. The XC track starts and finishes at the campground and is approx 8km's long. It is predomintaley single track with not a lot of elevation but a few obstacles thrown in to keep it interesting. You can also follow the Munda Biddi from here into Dwellingup.
They are the only places I can think of that won't have you pushing up big hills or having to force your way through oober soft gravel, especially this time of year, winter is much better.
You can find info on the Munda Biddi here. http://www.mundabiddi.org.au
Hope it helps,