The flat we live in isnt big enough but my partner doesnt earn enough for us to move to a bigger house, help?

I am a stay at home mum and dont get paid, well i get $95 a week for my son (which every family with kids gets if they earn under 100k) and we get an accommodation supplement of $95 a week to help with rent. Our rent at the moment for our tiny 2 bedroom unit (block of 4 units) is $265 a week which is just under half of my fiances weekly wage ($570 without any overtime). We also have 2 dogs which makes where we can move to very limited as landlords arent usually too pet friendly. I have been looking into moving into a 3 bedroom house but to get it in an ok area that isnt filled with crime and low lifes is about $350-450 a week. My partner often does overtime which can bring his weekly wage up to about $800-900 a week, but it is never guaranteed and there can be periods of months where there is no overtime at all. I know if we moved into a house with that much rent that the goverment would pay more towards our accommodation supplement but i still dont think it would be enough, especially if there is no overtime. We have a 13 month old son and another baby due in november, so desperately need the space but we cant seem to afford it. Does anyone have any ideas on what we could do? And no im not going to even think about getting rid of my dogs just so that rent is cheaper, that is not an option at all