I don't think you have an appreciation for how much fuel is required to adjust an asteroid's orbit to approach earth. Much less align with earth's orbit. Much less get it into stable orbit around earth.

Or the fuel required to get the fuel you need to the asteroid.

Fuel that has to be lifted into orbit. Which requires fuel.

Go look up the costs of lifting equipment just into orbit. Then look at how much lithium is worth per kg.

Then go look at the fact that we have exactly zero manufacturing capacity in space. Simple operations and repairs using pre-made parts on the ISS take hours-long spacewalks that require weeks of planning. And you think we can "build engines" on asteroids?

I was thinking that we could send a bunch of engines inside a starship to mount on the side of objets that we want to take back and we can fuel them with material that we find on the object we want or on near by objects.

Why not just use the engines that come with the starship? In any case, both shipping the asteroid bit by bit inside a rocket or converting the entire asteroid into a rocket will need basically the same amount of fuel. You would need to bring over the lithium ore processing equipment as well, so you only need to transport refined lithium metal back to earth. You might even be able to use the non-lithium parts of the asteroid as reaction mass in that case.

At $1.7m for 100t, it probably wouldn't even be worth sending a Starship to orbit to fetch it down.

I’m pretty sure you don’t do any of these things.