What a fun project!
I am not sure about picking them green, you may need to just give it a try and see. Mostly I have found it works best to let them mature on the plant.There is a technique called Embryo Rescue where they take the immature seed and put it in tissue culture to make it grow, but that sounds like a real science project!
My book by a fellow from the Royal Botanic Gardens at Kew says C. rotundifolia can be treated like a hardy annual, and started in a greenhouse or cold frame.They germinate quicker when warm, but will germinate when cool. He says they do not require a cold period.
Then there was a guy named Deno, a chemist whose hobby in retirement was to check seeds of various species to see what worked best. He says for this species, fresh seed liked to be sowed warm, then a cold period, some germinated when cold, then if he put them in warm more sprouted. If he sowed them at 40 degreesF they did not germinate. He found seed that had been stored dry for 6 months just wanted to be put at room temperature (70F).
Plant World Seeds, an amazing website, says to sow any time, but this species does germinate quicker at 15-20 C. but they sow in an unheated greenhouse and just let germination happen whenever.
So have fun, maybe try different things. It's a great hobby.
Also, this particular species has a remarkable distribution, includes Canada, most of the U.S., and Europe and northern Asia and Russia. So there may be local variability. You are so far north possibly your subspecies might prefer a cold spell.
If they need a cold spell to break dormancy, don't forget it needs to be moist. Dr. Deno standardized an easy method- moisten a paper towel, fold it twice, with seeds in the center, then put it in a non ziplock baggie, fold that over, and put it in the fridge. some folks fiddle around with brief periods in the freezer, and say for some tricky plants this really helps.
This would certainly be a plant I would try "Winter Sowing". This is an easy technique where you sow them in gallon milkjugs in early winter, put them out in the yard with the cap off, so they go through the natural cold cycling of you region, but a bit protected. Then they sprout in the spring.
And don't forget to report back how your project goes!
