Hi Tim.
That's a shame about your many constraints. Knowing about them, though, will make it possible to work around them.
My first thought when you mentioned that the grass must stay is that you might want to convert that lawn, subtly and slowly, to a polyculture lawn that fixes nitrogen. That will add fertility to your whole yard, over time. Even just some clover sown into it will improve the surrounding soil.
I would look at square-foot gardening and companion planting for your garden space. I find that approach particularly well suited to garden beds.
At the end of the day, just tend the soil. Make sure your soil life has everything it needs, and resources will be made available and moved around by that soil life.
I would make sure there's enough organic matter in the soil, and remedy the situation if there isn't. I would perhaps look into biochar, and if experimenting with that appeals to you, try that out to retain structure and microbe hotels in active soils.
I think it might be necessary to consider drip lines under a three inch mulch layer, into which you make craters to the soil surface, into which you plant. When seedlings reach sufficient height, the mulch can be crowded closer in.
I think you might also need to consider a canopy or cover that you suspend to provide a dappling of shade when the sun is at it's most relentless.
As to invasive tree roots, you could dig a trench between your trees' root zones and any infrastructure you want to protect, and put in a textile root barrier.
If I was in that position and had my current access to wood chips from urban arborists, I would backfill that trench with a mix of woodchips and the material removed, and inoculate the filled trench with a
compost extract. That would act as a soil life bioreactor, nurturing the soil life we want and moving it out of the nursery into the rest of the soil. It actually can work that way. I have done it, with frankly astonishing results.
Of course, you are in South Australia, and there are probably things you can't do, like anything that encourages poisonous anything to nest around your house. I don't even know if you could use mulch like I do without incurring some horrible infestation of something or other.
But with that caveat in mind (thanks, Captain Obvious), feed the soil life. That much can usually be done largely surreptitiously.
-CK
A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.
-Robert A. Heinlein