### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

IaC = CDK.

Attempting to connect an API-GW endpoint directly to EventBridge.

```
      const awsIntegration = new apigateway.AwsIntegration({
      service: 'events',
      action: 'PutEvents',
      integrationHttpMethod: 'POST',
      options: {
        credentialsRole: role,
        requestTemplates: {
          'application/json': requestTemplate,
        },
        passthroughBehavior: apigateway.PassthroughBehavior.NEVER,
        integrationResponses: [
          successResponse,
          serverErrorResponse,
          clientErrorResponse,
        ],
      },
    });
```

Receiving response:

```
    {
      statusCode: 400,
      data: {
        Type: 'User',
        message: `Error invoking integration for API Gateway ID '1z7ky5zzcb': API Gateway AWS integration action URI "arn:aws:apigateway:ap-southeast-2:events:action/PutEvents", method "GET" not yet implemented`,
        __type: 'InvalidRequest'
      }
    }
```

Similar to bug reported:

* https://github.com/localstack/localstack/issues/5508

### Expected Behavior

* 200 response
* Event created in relevant Eventbus

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

docker run localstack/localstack

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

Calling the API endpoint using axios from jest tests.

### Environment

```markdown
- OS: MacOS
- LocalStack: latest
```


### Anything else?

_No response_
)
        snapshot.add_transformer(snapshot.transform.regex(target_id, "second-invalid-target-id"))
        snapshot.match("length_error", e.value.response)
        target_id = f"test-With_valid.Characters-{short_uid()}"
        aws_client.events.put_targets(
            Rule=rule_name,
            Targets=[
                {"Id": target_id, "Arn": queue_arn, "InputPath": "$.detail"},
            ],
        )
    @pytest.mark.aws_validated
    def test_should_ignore_schedules_for_put_event(
        self, create_lambda_function, cleanups, aws_client
    ):
        """Regression test for https://github.com/localstack/localstack/issues/7847"""
        fn_name = f"test-event-fn-{short_uid()}"
        create_lambda_function(
            func_name=fn_name,
            handler_file=TEST_LAMBDA_PYTHON_ECHO,
            runtime=Runtime.python3_9,
            client=aws_client.awslambda,
        )
        aws_client.awslambda.add_permission(
            FunctionName=fn_name,
            StatementId="AllowFnInvokeStatement",
            Action="lambda:InvokeFunction",
            Principal="events.amazonaws.com",
        )
        fn_arn = aws_client.awslambda.get_function(FunctionName=fn_name)["Configuration"][
            "FunctionArn"