### 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_
]
        aws_client.events.put_rule(
            Name="ScheduledLambda", ScheduleExpression="rate(1 minute)"
        )  
        cleanups.append(lambda: aws_client.events.delete_rule(Name="ScheduledLambda"))
        aws_client.events.put_targets(
            Rule="ScheduledLambda", Targets=[{"Id": "calllambda1", "Arn": fn_arn}]
        )
        cleanups.append(
            lambda: aws_client.events.remove_targets(Rule="ScheduledLambda", Ids=["calllambda1"])
        )
        aws_client.events.put_events(
            Entries=[
                {
                    "Source": "MySource",
                    "DetailType": "CustomType",
                    "Detail": json.dumps({"message": "manually invoked"}),
                }
            ]
        )
        def check_invocation():
            events_after = aws_client.logs.filter_log_events(logGroupName=f"/aws/lambda/{fn_name}")
            
            assert len([e for e in events_after["events"] if "START" in e["message"]]) >= 1
            assert (
                len([e for e in events_after["events"] if "manually invoked" in e["message"]]) == 0
            )
        retry(check_invocation, sleep=5, retries=15)
    @pytest.mark.aws_validated
    def test_put_events_nonexistent_event_bus(