Hi and welcome, Renee. Could you possibly post a picture of your tree and the area under it? I haven't been able to find anything about "Hawaiian pepper tree" either in our database or other online sources, so I'm wondering if your tree could be a Brazilian pepper? It's (unfortunately) a very common invasive tree here in Florida.
In general a large tree with a shallow root system will absolutely deplete the soil under its canopy, just because its roots are using up all the available water and nutrients. I think you're going to have trouble growing grass under there unless you are willing to constantly feed and water it. Even then, the tree is going to get a lot of the goodies meant for the grass and will grow more canopy and roots, too. The tree's shade is another issue for growing grass because most lawn grasses like more sun than your average tree is going to allow.
You might be better to try mulching the area under the tree and growing something like bromeliads - they love growing in the shade of trees and don't need much in the way of nutrients from the soil. I also have a beautiful groundcover under part of my mango tree - Texas longleaf jasmine. It will grow with little or no maintenance in filtered shade.
