Say I design a two-sided PCB. I want the bottom side to have just a 2xAAA battery holder. Could a pick and place machine handle this or is it out of the question? What happens when the top side is flipped over - wouldn't the board "wobble" depending on the height of the components on the top side?
If you find a surface mount AAA holder, then typically you'd install all the parts on the other side, reflow it, then flip it, solder stencil, place, and reflow the battery side.
You don't have to worry about wobble, the circuit boards are handled by their edges. The parts on the bottom will stay on due to surface tension, but if you do have any large or heavy parts then the assembly process can include glue to secure them to the board.
Large assembly runs more frequently involve having the batteries taken care of by the case, with sprung contact touching pads ont he PCB once assembled. If the batteries must be mounted on the PCB, throughhole mounting by hand isn't that expensive per board. If you can design it so it'll go through a solder bath it'll be cheaper, but there is machinery available that does automate soldering through hole components as well so you can automate everything if the volume is high enough.
You should talk with your assembler. It seems that you might not have much experience with designing for manufacture, and you might be making assumptions that aren't necessarily true. Yes, through hole hand assembly is more expensive than surface mount, but you might be surprised at how affordable it is, particularly once you get to high volumes.