How to fill 4" width gap between the walls and alcove bathtub door
This isn't the only way to approach this, but since many shower doors have a U shaped channel that covers another U shaped channel (that's been attached to the wall), you could use box section aluminum -- 2"x1" (assuming that the 1" dimension meshes well with your chosen door) on both sides. Your local metal supplier probably won't have this in stock, but you could order from them or the internet generally.

Tip: drill a weep hole at the bottom of the 2x1 that drains to the tub so that you don't inadvertently get standing water in the tube.