This may be worth a try,
www.greendots.com.sg (not sure if there is the .sg), its run by Dr. Vinisha (9010 5004), she uses her home in a black and white for science modules for kids and sometimes she runs science based b'day parties. I know your kids may be a bit young for her bday parties, but if its a weekend where she is not booked, she may allow you or rent you the spot?
Her place has nice big trees, climbing structures, she can take out the pool, kids can run around, a resident retriever....
We go often just to hang out, my kids test out her science modules and its just nice to have a big area to run, definitely picnic space.