Lithium is light. This means that it generally floats to the top of things. This means that it is not a good candidate for asteroid mining, because while it has floated to the surface of the Earth over the course of many many many years, it has not floated to the surface of asteroids.

This is also why things like gold or platinum are good to mine from asteroids, because they have not had the opportunity to sink to the core of those asteroids like they did on Earth. (And getting to the core of an asteroid is a significantly more reasonable proposition than getting to the core of the earth)

Lithium is $17,000 per tonne. Even if you found a pure Lithium asteroid, it wouldn't be worth shipping in a 100 tonne Starship that would need a dozen refueling trips to make it to the asteroid and back

You don't use something like Starship to haul lithium back from the asteroid belt 100 tonnes at a time.

You use something like Starship to deliver construction equipment that can be used to build engines on the asteroid it to earth orbit for processing.

Ideally you'd bring the minimum amount of equipment necessary to do in-situ production of the rest of the equipment, but we're not at that boot-strap self-replicating level quite yet.

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.