Your biggest problem will be soil too warm after sprouting , this will make seedling too tall/leggy. Long ago I grew many plants from seed in Chicago area. Once sprouted , I moved the flats to a garage that was usually 50 to 60 F. I used 48" shop lights with a mix of grolux and white bulbs but I added 2 incandescent sockets to each hood with 60 watt bulbs ( ...
I ended up getting four 48" double bulb shop lights. I started with 10 flats on heat mats. Transplanted into 3 or 4 inch pots, then 6 inch, and hardened them off outside about a week before I planted them out. It didn't seem to mater which bulbs I used. I tried different kinds the next summer. It Worked great.
This website confirms it is a "Black Nightshade"
They grow green or deep purple berries which are poisonous to most animals.
Leaves are also poisonous.
If you have any pet I recommend to pull it out with gloves or use a weed killer.
This is not a tomato plant.
Although if my assessment is correct, it is of the same nightshade family Solanaceae.
I agree with your assessment black or blackberry nightshade (Solanum nigrum) or as we sometimes know it in Australia... deadly nightshade, due to its toxic effect when grazed by livestock.
The leaves and fruit are also toxic to humans.