🐛 BUG: A request to the Cloudflare API (/memberships) failed.

### Which Cloudflare product(s) does this pertain to?

Workers

### What versions are you using?

3.109.2 [Wrangler], 20.18.3 [Node]

### What operating system and version are you using?

Mac Sequoia 15.3.1, Docker dev container

### Please provide a link to a minimal reproduction

_No response_

### Describe the Bug

I'm getting started with Cloudflare. So far, I've run `wrangler init` to create a backend and have set the CLOUDFLARE_API_TOKEN environment variable to my API Token. I've given that token as many edit permissions as possible.

When running `wrangler deploy`, I get the following error without any further details:

⛅️ wrangler 3.109.2
--------------------

✘ [ERROR] A request to the Cloudflare API (/memberships) failed.

Unable to authenticate request [code: 10001]

If you think this is a bug, please open an issue at:
https://github.com/cloudflare/workers-sdk/issues/new/choose

### Please provide any relevant error logs

_No response_

Start at Wrangler's deploy flow for the /memberships API request and reproduce with Wrangler 3.109.2, Node 20.18.3, and the supplied API token setup. Trace the authentication failure; done means identifying the cause and documenting or implementing a verified resolution for valid Workers API tokens.

Written by the indexing model from the issue text.