I'm thinking something like a robot with treads, that can move in a straight line (or maybe curves), perhaps at the end of a long boom with a conveyor on it that moves bricks from a central source.

It basically grabs a brick off the end, squirts some mortar down, and lays the brick. Make the bricks shaped in a special way so they interlock better and need less mortar at the edges (they already have bricks something like this; they almost look like lego). Then it rolls to the next brick position.

Once it gets to a certain point, it "crawls up" to the next level, perhaps by gripping the level below with the treads (the mortar will probably need to be some kind of "ultra-fast setting" stuff; possibly more like epoxy than mortar), and lays the next layer, being guided by the last.

I guess I'm just saying this particular brick laying robot looks like it was designed by an engineer who, back at the beginning of heavier-than-air flight, thought an airplane needed to flap its wings to get off the ground...

Or better yet: imagine the smallest walking/climbing platform that could carry a robotic arm just big enough to take a brick and pass it on, then coordinate many of them into semi-static bucket brigades for bricks and mortar that could form an ad-hoc power grid to solve the battery issue.

The main innovation tasks/requirements would be cheap volume production of the swarm elements and intelligent planner software to automatically derive a work plan straight from the CAD data (or else project setup/customization would be prohibitively expensive). The beauty would be scaling: need to build a bigger structure? Add swarm elements. Need to speed up the schedule? Add swarm elements.