Add mulch under the tree instead of lawn.

Lawn always struggles when less than 6 hours sun daily.

It is always difficult to establish and maintain a healthy lawn under the canopy of any large growing tree.......it really doesn't matter much about the species. The wide spreading root system will tend to outcompete the lawn for moisture and nutrients and the canopy will prevent good sunlight reaching the lawn. As noted above, it makes more sense to replace the struggling lawn with mulch. Better for the tree as well.