I would leave it alone. The tree is currently water saving, self mulching and maintenance free. You don't need to remove the debris it drops. As you say, the ground absorbs it. Why make work and why make ugly with fake turf and artificial rocks? You could keep the area around the tree neater if you want by delineating a clear circle around it using a slit trench edge. Otherwise leave it be.

If this is your only spot for a useable lawn area, it sounds like it could work. To be fair, I don't have artificial turf, but Eucalyptus seed pods seem like they would be too big to embed into the turf. They are acorn size.

I found this:

How to maintain artificial turf with eucalyptus treesRegular and diligent maintenance is the key to preventing problems caused by eucalyptus pods and other debris.

• Remove debris regularly: Use a leaf blower to clear loose leaves and pods from the surface of the turf. For stubborn or embedded debris, a stiff-bristled broom or a plastic rake is effective. Avoid using metal rakes, as these can damage the turf fibers.
• Rinse the turf: Hose down the area periodically with a garden hose to wash away smaller, loose particles of dirt and debris. This is especially helpful after heavy shedding. Avoid using a high-pressure washer, which can damage the turf or displace the infill.
• Brush the fibers: Use a push broom or a turf rake to brush against the grain of the fibers. This will help them stand upright, loosening any embedded debris and redistributing the infill.

We have rock mulch and a leaf blower works to clear debris so long as it weighs less than the rocks. One inch lava rock would be quite light so make sure you will be able to move the debris without moving the mulch with your blower.