Define what space is lawn and what is flower bed. Repair the lawn. Mulch the bed. Water as needed.

Do you think overseeing is sufficient?

Mulch or new soil? I plan to replant the area.

I'm fine with grass seed. I also don't generally give a crap about the lawn in a garden.

Unless you are planning on changing the grade further or bringing the excavator back in and removing soil to replace it, the soil you got, is what you got. Any organic matter you may want to add, be it manure, compost or mulch is not new soil. It is organic amendments that should just be spread on top of the space that is bed.

It’s a lawn beside a garden. Not a lawn in a garden. This is our entire backyard.

It’s a lawn beside a garden. Not a lawn in a garden.

Yes, of course. Define what space is lawn and what is flower bed. I was referring to my general feelings about the lawn part of a garden. It's just something green to walk on to get from one part of the garden to another. Other than that I could care less about lawns or what they look like.

Gotcha.

For us as a family of five with a dog the lawn is important and well used

Then it just needs to be green and cover the dirt and all is good.