I agree they will be fine in your little house with a well secured heat lamp. Make sure the heat lamp is rated for outdoors and cannot be knocked off into the straw or bedding as it can (will) cause a fire (I lost a hen coop to that).
To make you feel better, when my hen brooded her chicks in February, the chicks were running around without her covering in a matter of days. Then they run to huddle for a while, then they run around in the cold again. It's being a normal chicken. You'll recreate that in the hen coop with the heat lamp. They don't need to be inside, and are actually better outside. They'll grow and feather faster.
Lady of McCamley