The scenario I was using was using excess power that cannot be...

The scenario I was using was using excess power that cannot be fed back into the grid and of course a large battery to be able to store that excess power for nighttime use. Solar systems only allow you to feed so much back into the grid, so if the excess capacity is there that cannot be used the system shuts down. You can't sell potentially all the power you can generate back to your energy provider.

If power goes up as you say it will with the price of fuel, although I don't think the two are as closely related as you think, then this will move more people onto more solar power generation and storage, hence the need for more lithium in the world anyway.

Either way, the cost of power and or the cost of fuel going up, is likely to contribute to more consumption of lithium and other critical minerals.