Do you know much about the ecology of the area you live in? I'm at a loss when it comes to Australia, other than there's Eucalyptus forests that require natural forest fires to regenerate. So I'll suggest the principles of my idea and if you think it's worth considering you can let me know if you know/can get hold of the information. Or maybe someone else on the forum will have some expertise.
First of all, find out what is likely to naturally colonise her lawn without the spraying. Then find out what animals it would attract. It might also be worth seeing if there are any plants that attract particular birds/animals she may like that would be happy in the conditions, even if they wouldn't colonise naturally (natural colonisers would be best though).
Then wait for opportunity. A few ideas might be if her dog likes chasing birds (so she might like to get more birds in her garden to give the dog some excercise or something) or if you can find out if she is particularly fond of any of the plants/creatures that might naturally colonise her garden.
As soon as you can find a hint of an oportunity, either plant some things to attract said creatures to your garden (then hopefully show her how they like your garden and if she shows an interest you can explain why they like your garden and how easy it would be for her to make an equally attractive area for them in her garden, maybe along your boundary
) or you could get her a plant or two as a present and show her some pictures or somehting of the nice little creatures they could attract if she decides to give them a chance.
But don't mention the spraying. Just try to get her interested in the alternatives, if and when opportunities arrise. Hopefully she'll change her mind about putting the effort into spraying when she could have a more interesting garden than an expanse of lawn or she might at least consider shrinking the lawn a bit.
That's my only idea at the moment anyhow.