Hey TJ, hadn't see you around in a bit.
Maybe grass all the way under the tree, up to the trunk isn't the best idea. Well, actually its not. However, I suspect that it seems unsightly. The tree and grass are in direct competition with one another for soil resources (water, nutrients. don't know if grass utilizes beneficial fungi that fix nitrogen).
Perhaps a better combination is a good circle of organic mulch, a lower maintenance edging,then grass outside that. Possible some small understory plants to add aesthetic value.
The mulch will help in many ways. Slowly enriching the soil as it breaks down, buffering the temperature by insulating, retaining soil moisture, reducing soil compaction that comes with mowing, reducing the likelihood of lawn care equipment damage to the trunk of the tree (mower bumping/ string trimmer whipping), avoiding/ reducing the competition so the tree can be healthier, and the lawn can be healthier.
If you could convince them of this, then use the dripline, or as large of an area as possible, as the mulch ring guide. And easy to maintain edging of some sorts helps to keep the mulch in place and tidy looking.
Scalp the grass that will be under the mulch with a mower and then string trimmer. Layer overlapping newspaper several pages thick, or overlapping cardboard to help to smoother the grass and weeds. Lay organic (wood chips, ground bark (trying to avoid cypress mulch, some of which is harvested from some gulf coast wetlands, destroying them in the process)) for a final layer 3-4" thick, while staying a few inches away from the tree's trunk.
Consider if plug aeration is needed. Overseed the lawn. Keep it watered during establishment.
If needed, thin the canopy for light. You can probably find a shade tolerant grass seed, if light really is a factor.