How do I use FormKit inside a Vue based web component

Question

See this sandbox: https://codesandbox.io/p/devbox/little-breeze-lnlfzp

• I've defined the two components as custom elements
• Registered them both in main.ts
• Created a Vue App with FormKit as plugin and added it to a random div defined in index.html
• The two components work ok and communicate over the pure js main site.
• When adding FormKit elements to the component I get warnings and errors and nothing is shown.

My questions:

1. How do I get the FormKit elements to work inside the custom element? I've added it to the first component and I get an error, even with using FormKitRoot. Maybe I'm using it wrong?
2. How do I prevent vite from including Vue in the web components and make them share the existing app instance?