Attach AWS API Gateway Custom Authorizer to a Route

Question

I would like to attach a custom Authorizer to an API Gateway Route.

The API Gateway and the custom Authorizer are deployed with Terraform together with the whole infrastrucuture.

Now I want to deploy some Lambda Function as Routes of my API Gateway.

I first tried to deploy the Lambda Function with Serverless Framework, but apparently it is not possibile to attach a pre-existing Authorizer.

In the documentation they show this example:

provider:
  name: aws
  httpApi:
    authorizers:
      customAuthorizer:
        type: request
        functionArn: arn:aws:lambda:us-east-1:11111111111:function:external-authorizer
 
functions:
  hello:
    handler: handler.hello
    events:
      - httpApi:
          method: get
          path: /hello
          authorizer:
            name: customAuthorizer

But this will create a new API Gateway.

So I added the ID of the Gateway.

provider:
  name: aws
  httpApi:
    id: xxxxxx

But then I get:

Error:
Cannot setup authorizers for externally configured HTTP API

Apparently Serverless don't allow to setup an authorizer for a pre-existing HTTP API Gateway.

So I decided to use the AWS CLI.

I run:

aws apigatewayv2 update-route --api-id <gateway-id> --route-id <route-id> --authorizer-id <authorizer-id> --authorization-type custom

but here I get the error:

An error occurred (BadRequestException) when calling the UpdateRoute operation: Unable to update route. Authorizer type is invalid or null.

event though the ID is correct and pointing to the Authorizer.

Final question

Does anybody know how to attach a custom Authorizer to a Route in API Gateway, where the Authorizer and the API Gateway are already deployed?