1 - Set stage variables for all your environments on API GW, in name use "cognito_pool_id" (or whatever you prefer) and in value add your cognito pool id for each environment.
2A - In case you already have your Test environment with authorizers all placed in your methods (under your api resources), just go to your current authorizer settings (in case you're already using a cognito authorizer for your methods) and change the static ARN for your stage variable
2B - In case your methods still don't have an authorizer using cognito or you're using a lambda authorizer, well, you gotta create a new authorizer (as we can't change the authorizer type) and set it up for all your methods (I hope you don't have too many methods because it consumes some time).
Here's an example of the ARN using stage var: arn:aws:cognito-idp:your-region:111222333444:userpool/${stageVariables.cognito_pool_id}