I wouldn't use either the dome or the heat. Those seeds just don't need it.
Just sow the seeds inside, then immediately they sprout put them somewhere where they have bright light but without direct sunlight(which will cook them when young). As they grow taller move them to cooler, brighter areas & then on to a sheltered position outside (to gradually harden them off) -- lastly transplant into final position.
Hattie-great advice. I love the time difference ( it is 1:00 in the am here). I can always count on my British friend to answer my dumb questions in the middle of my night!
OK-I will DC the whole dome thing. I might leave the heating pad on very low, as the seeds are in a guest room that stays quite cold (we keep the door shut to help with the utilities). When the seeds sprout, I will leave the door open so they won't freeze. The are near a south facing window, so that should be ok (?). Ill trasplant them (when they get bigger) into peat pots, then harden them before planting outside.
Not a UK friend, more like one time zone over. (Did you know that southern Idaho is on MT but northern Idaho is on PT? By the time I hammer this out . . . it will be 5am here)
What you plan looks good to me, DG.
Don't worry too much about them freezing indoors. Those cabbage-family plants can take the cool temps.
I believe it was Wifezilla who said that "modern" windows don't have the "cooking" risk as other windows. Don't know about that - my best south window is on the way out with a modern replacement ordered, so I guess I'll find out. It is already a double pane and there's a low angle to the sun . . . and that sneaky Pacific NorthWest sky sneaks in here during late winter/early spring with lots of clouds.
Maybe just a short debut in the sun the 1st day then move them out for a few hours during mid-day the 2nd. You've got that bright, New Mexican sun .