Accessing Pyramid Settings throughout the program
Pyramid itself does not use global variables, which is what you are asking for when you ask for settings to be available in class-level or module-level attributes. For instance-level stuff, you can just pass the settings from Pyramid into the instance either from the view or from the config.
To get around this, you can always pass data into your models at config-time for your Pyramid app. For example, in your main just pull settings = config.get_settings() and pass some of them to where they need to be. As a general rule, you want to try to pass things around at config-time once, instead of from the view layer all the time.
Finally, a good way to do that without using class-level or module-level attributes is to register instances of your services with your app. pyramid_services library provides one approach to this, but the idea is basically to instantiate an instance of a service for your app, add it to your pyramid registry config.registry.foo = ... and when you do that you can pass in the settings. Later in your view code you can grab the service from there using request.registry.foo and it's already setup for you!
I have a pyramid API which has basically three layers. 

View -> validates the request and response
Controller -> Does business logic and retrieves things from the DB.
Services -> Makes calls to external third party services. 

The services are a class for each external API which will have things like authentication data. This should be a class attribute as it does not change per instance. However, I cannot work out how to make it a class attribute. 
Instead I extract the settings in the view request.registry.settings pass it to the controller which then passes it down in the init() for the service. This seems unnecessary. 
Obviously I could hard code them in code but that's an awful idea. 
Is there a better way?