What sp of Lespedeza you have? Where in Ontario do you live?

Probably thunbergii. I am in zone 5b near Brockville. The parent plant is gorgeous!

I don't know what Lespedeza thunbergii chilling requirements are, but judging by its hardiness zone (4-9) I would think it needs some time in winter with air temperature below 0 C. I am not sure how it translates to soil temperature which is relatively higher but for sure not warm. Since Lespedeza thunbergii branches normally die in winter it is the soil temperature that matters.

From your three options I would choose the first one and put it in the cold storage. I believe it needs dormancy period.

Personally I don't have any experience with growing Lespedeza thunbergii though and hope that someone with experience will chime in to give you more informed advice.

I am thinking, may be I would even bury for winter the pot it is in in the ground outside and, taking into account its vulnerability, give it some extra protection in the form of mulching.

Thanks, Windberry, I wish two cuttings had survived so that I could try both options. What I might do is in between both - sink it in a bigger pot and keep it in my workshop, which is just at or above freezing, with my hydrangea macrophyllas. Unless someone else has had success with something else. We have had a series of winters with a lot of freeze-thaw cycles so, even under mulch, is not very secure.