Try making API calls from the content page to avoid the cloudflare human checks

[meooow25]

Suggested by @RimuruChan in https://github.com/meooow25/carrot/issues/56#issuecomment-2241206541

It makes no sense to check that an API request is made by a human, so really we shouldn't need to do this. But CF doesn't seem to care so I have no hope on that front. This might just get the extension working reliably again.