Hey Jordan,
little sidenote - that token won't just automatically pull from the .env file - you have to set your environment up to manage them, I think this is the right way in nuxt.
https://github.com/nuxt-community/dotenv-module
Otherwise - I'm not seeing any GSAP specific errors in that netlify log. Your issue isn't a 403 with the private repo, the issue is here -
Cannot find module @rollup/rollup-.... npm has a bug related to optional dependencies (https://github.com/npm/cli/issues/4828). Please try npm i again after removing both package-lock.json and node_modules directory
Here's a thread. Sorry I can't help more here!
https://github.com/nuxt/nuxt/issues/25006