You don't need grass. If it provides too much shade, you can thin the canopy.

It looks like they use the lawn as a play area. I agree thin the canopy. I found this:

For the seating area and around the base:

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Are you willing to give up the shade it provides in exchange for grass? I wouldn't, but I don't know how hot it gets in your area in the summer.

There is NO type of turf grass that will grow well in much shade. Forget what Google or other sources may tell you. And even with less shade, a lawn will not grow well under the canopy or dripline of a large tree, Tree roots will always outcompete the grass for water and nutrients.

Look for drought and shade tolerant groundcovers to replace the lawn. Or extend a mulched area around the base of the trunk, then plant the groundcovers or other larger plants that share similar attributes.