Having grown up in Liverpool myself, my brothers and I attended Liverpool West Public school from K-6 which is located on the corner of Flowerdale Road & Hoxton Park Road.
We found it was quite a good multicultural school offering most of the facilities that the private schools in the area offer.
Not too sure what it is like now but back in our day it was one of the best public schools to attend.
Hope this helps
I don't know about Liverpool public school but is Leichhardt Public School an option?