Summary Checklist:
Keep your current token (the one with all the permissions). This is your User Access Token.
In your script, make a GET call to https://graph.facebook.com/me/accounts using that User Token.
Parse the JSON response to find the Page you want to post to and extract its unique access_token.
Use that new Page Access Token and the /{page_id}/videos endpoint to make your final upload POST request.
This two-step process is the standard, secure way to post content to Facebook Pages via the API and is the most likely solution to the permission error you're seeing.