These kind of questions is why you need to bring in a professional. This is a relatively simple thing to do; however, the assumptions you have made and your attempts are all leading to likely burning up a transformer, actuator and possibly more. If space limitations are such then one could mount the transformer in an external j-box and pull wire, or better yet get an actuator that is rated at 120VAC and get rid of the transformer. If your fan is so oversized I would look at putting an external switch anyway just to open the damper so you don't have to run the fan just to cook something little. *Just realized downdraft, so no fan then draft will likely go the wrong way, but* Then one could interlock an end switch and.....bring in a professional.
We don't want a conversation such as:
your insurance agent "Well it seems the electrical in your kitchen ventilator caused the house fire and since it wasn't inspected/proof that it was professionally wired your insurance policy is null and void on the damages caused.."
You to your agent "Well I learned how to do this on the internet at HVAC-Talk and CM, ...told me how!"
Good luck to you and you might post your general location, who knows maybe one of us lives near you/knows someone that does.