Right after posting this (and after a few hours of recapping basics in Dash), I realized I needed to pass in the current State. So this is how I resolved it:
Output('test', 'children'),
[Input('output-figure', 'clickData')],
#### ADDED HERE
[State('output-figure', 'figure')])
def display_click_data(click_data, figure):
if click_data is not None:
curve_number = click_data['points'][0]['curveNumber']
x_value = click_data['points'][0]['x']
trace_name = figure['data'][curve_number]['name']