He should go for 50/50. Which is probably 50% more than she wants to give him.

If he is starting this custody battle because the child told him she'd like to live with him, he's doomed. For of course the child told him she'd like to live with him when she was with him. But, at age six, she's probably made similar remarks to mom.
Or she told him this when she was in trouble with mom and didn't want to go home to face the music.

Mom may be a liar. She may be a gigantic liar, but unless her lying endangers the child, the court won't care.

No birth parent is going to want a schedule that says one parent can change it weekly depending on their work schedule. That's fair to no one but the guy who works the changing schedule. It leaves the BM wondering from week to week what days she needs to plan child care for. It leaves the child in a never ending will I or won't I see Dad today.

If this child is already in school and he doesn't live in the district, even a 50 50 is problematic.

I'd say tell him what he is up against, but I doubt he'd believe you. It sounds like this will be a costly lesson for him.

The only thing I might do is to tell him to ask his attorney what the chance of success is.

_______

While I wrote this the distance factor was added.

He won't get custody unless BM gives it up. Most he is going to get is ordered holiday and summer.
Who moved away will factor into transportation costs.

He really needs a court order so going to court is necessary.
But, he should not get his hopes up for full custody.