Ultimately I am going to have a dedicated small form-factor machine permanently setup on my TV. While I'm still in the very early stages of this, and streaming services try very, very hard to make this as difficult as possible (and should, in general, go fuck themselves), I put some of the initial important information here.
You will need some of the following if you have a simple laptop.
pacmd list-cards | grep outputLists available audio output modes.
Then
pactl set-card-profile 0 output:hdmi-stereowill switch the audio output to the default HDMI output.
pactl set-card-profile 0 output:analog-stereowill switch back to internal speakers.