Carrot Pie: custard or stew?
I can see this as being similar to a sweet potato pie.  Best bet in that case would be to cook the carrots, then puree and mix with eggs, milk, etc. just as if it were a sweet potato or pumpkin pie.

Two alternatives that come to mind would be to treat it like a fruit pie, as you say.  Because of the texture of the carrots, I would grate them with the large holes on a box grater, toss with sugar, raisins, some tapioca and whatever spices your heart leads you toward.

An alternative would be to make a molded gelatin like pie, similar to a refrigerator lime pie.  Lots of gelatin and grated carrots.  I, personally, wouldn't like it, but I am not fond of jello with carrots in it either.

Happy experimenting.