You're encountering this error because the Spree API extends ActionController::Metal instead of ActionController::Base.
The airbrake gem performs some logic shown here: https://github.com/airbrake/airbrake/blob/68543097a8731cf1f3e717946bd5ed33e5edccad/lib/airbrake/rails.rb#L13
which includes methods in to ActionController::Base.
You have a few options on how to get around this:
- Load the methods in to ActionController::Metal as well using a similar method to that shown in the airbrake gem
- Load the methods in to Spree::Api::BaseController using a similar method
- Decorate Spree::Api::BaseController to include the methods directly
That should get you going.