Fiddler actually works via a proxy, and automatically sets proxy settings in IE when running. You can check this by launching fiddler and then checking "Lan Settings" under Internet Options > Connection.

Invoke-RestMethod has a -Proxy parameter you can use to point to Fiddler's proxy. I think the default is 127.0.0.1:8888 :

Invoke-RestMethod -Proxy 'http://127.0.0.1:8888' ....

Edit: Adding screenshot to confirm the answer above.