I don't understand why moving an item, like a bed or even a sculpture requires construction.
Should it not be under hauling?
You try to haul those two person beds through the small one-cell wide doors in this game ;) You first have to partially deconstruct it, then haul the package of planks and reconstruct it in the other room. So, actually, it should require both construction and hauling... and if a pawn can't haul but can construct and another can't construct but haul, you should need both of them to complete the task.
Construction is a superset of other tasks. There is a mod that enables haulers to do the hauling subset of construction jobs. This should include moving furniture. Hand me that brick (https://ludeon.com/forums/index.php?topic=26077.0)