I am an electrician. If it's straight down and there is an actual existing hole it should be pretty easy to fish in some more cables without doing any damage to the wall if there is no insulation.
You could always improvise by first fishing in a string with a weight at the end or use some sort of wire that is a little bit more ridgid and then use it as a pull string to tie your hdmi cable onto after you get from hole to hole.
