OK I figured it out.
For anyone in the future looking to use Construct 3 and Spriter Character Maps (where you can swap out a sword, body part, or clothing on an animated character - here's what you need to do...)
1) Follow the guide in the video I posted - this covers the Spriter part, and is still pretty accurate.
2) Exporting is the same process as it is for Spriter to Construct 3 without character maps. You don't have to export the Character Map file, or include it in the ZIP you'll be creating.
3) Import it into Construct 3 as normal (drag the .zip onto the layout).
4) To swap out a part in Construct 3, use the action Append Character Map on the spriter object you've imported. The value you give is the name of the character map you created.
Eg.
Will swap to this character map in Spriter: