No connection to a lake is necessary, imagine 2 locks in parallel both connected to the same low water level canal section "on the left" and both connected to the same high water level canal to the right (its not necessary to have 2 locks in parallel, its just easier to demonstrate the scheme doesn't require significant energy consumption):

Imagine one lock is currently filled to the high level, while the other is currently filled to the low level. The locks have been closed. As you let water from the high lock flow into the low lock through Archimedes screws, those screws can lift a weight, until they have equilibrated at the midpoint of the high and low water levels. But you have raised the weight and can now use that to power the draining of the lock that has only halfway sunk in level, by pumping its water to the lock that has only halfway risen. Apart from friction there is no energy consumption. Slow quasistatic transfer drives the friction energy consumption to 0. In practice you want to use an "extra" weight to attain higher pump speeds, and then recover the kinetic energy. That will consume more friction energy, but a small number of solar panels suffice, the energy is not used to lift water, but to overcome friction (haste makes waste).

Surely figuring this out would be a very marginal increase in time spent on writing your large post and series of responses.