Need card-mod help: how to reach <button-card>

I need to set flex: 1 1 auto on 2 of my custom button card instances, but I can’t figure out how to reach the <button-card> element. I have tried from the horizontal-stack and I have tried from the button-card. I’m unclear from reading the documentation whether this isn’t possible with card-mod or if I’m just not understanding how to define the style. Does anyone have any tips?

Example DOM:

Example yaml: (I removed the styles that weren’t working to simplify the starting point)

  - type: horizontal-stack
    cards:
      - type: 'custom:button-card'
        template: container
        color: '#1F6B25'
        name: Office
      - type: 'custom:button-card'
        template: container
        name: Game
        color: '#F4BC78'