Just an update: We decided to just drop it down an A/C duct (good suggestion!). We tied a heavy screw onto the end of a string and chucked it down the shaft, then drilled through the side of the duct to make an access hole, and drew the Ethernet cable back up through it. We patched the hole in the duct with....DUCT TAPE!
First time I've ever used duct tape on ducts. So yeah, it's all working now! Good call on everyone who told me to not use the coaxial -- that would've definately been more difficult.