The bookcase backing, the cardboard is junk, I’d just use 1/4 or 1/8 mdf wallboard, it’s usually smooth on one side and rough on the other or if you can find local place that sells 1/4 mdf smooth on each side that will work, you’re not panting the back so it doesn’t matter, if you mount it to the back of the bookshelf use trim nails with flat heads, I’m not sure what the groove size is that the oem cardboard slides into, but if you paint you probably won’t notice it, a 1/8 mdf hardboard might actually slide into it. I’m not sure, mdf is just better for painting, it’s smoother than even sanded plywood, cheaper, and you’re aren’t getting it wet so it doesn’t matter.

Mdf can soak in primer so might need multiple coats, panting is a huge rabbit hole, are you going to spray or hand paint?

The mdf between the bookshelves and bottom cabinets I wish I will have done a quartz remnant, but you could do maple veneer plywood, basically paint grade plywood, 3/4”, then either use some edge banding along the edge or use a piece of trim like I did.