### 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_
if "Name" not in kwargs:
            kwargs["Name"] = f"rule-{short_uid()}"
        resp = aws_client.events.put_rule(**kwargs)
        rules.append((kwargs["Name"], kwargs.get("EventBusName", "default")))
        return resp
    yield _factory
    for rule, event_bus_name in rules:
        targets_response = aws_client.events.list_targets_by_rule(
            Rule=rule, EventBusName=event_bus_name
        )
        if targets := targets_response["Targets"]:
            targets_ids = [target["Id"] for target in targets]
            aws_client.events.remove_targets(
                Rule=rule, EventBusName=event_bus_name, Ids=targets_ids
            )
        aws_client.events.delete_rule(Name=rule, EventBusName=event_bus_name)
@pytest.fixture
def events_create_event_bus(aws_client):
    event_buses = []
    def _factory(**kwargs):
        if "Name" not in kwargs:
            kwargs["Name"] = f"event-bus-{short_uid()}"
        resp = aws_client.events.create_event_bus(**kwargs)
        event_buses.append(kwargs["Name"])
        return resp
    yield _factory
    for event_bus in event_buses:
        aws_client.events.delete_event_bus(Name=event_bus)
class TestEvents:
    def assert_valid_event(self, event):