Leave it down, under the water, otherwise snow, rain & ice forms around the sliders and supports that will expand ten percent when it changes from water to ice and that can maybe mess up the lift.
If one water circulator doesn't make a big enough circle of open water and there's still a corner that freezes up, go with the Lowe's 1/6-hp utility pump for $65 as an add-on pony circulator, as opposed to adding another $450 circulator.
The legal seasonality issue on boat lifts is probably never-ever enforced, so it's not a problem!
What can be a big problem is all the noise created by a circulator if there's a year round resident living next door. The circulators definately do a good job of disrupting the quiet of a winter day or night with long running, noisy splish-splash surface water, and your good guy, quiet unassuming neighbor may be driven over the edge, get real pissed off, and just go pull the plug on your circulator when it's 10 degrees-below and you are away.