### 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_
moto_backend = events_backends[store._account_id][store._region_name]
            event_bus_name = get_event_bus_name(event_bus_name_or_arn)
            event_bus = moto_backend.event_buses[event_bus_name]
            rule = event_bus.rules.get(rule_name)
            if not rule:
                LOG.info("Unable to find rule `%s` for event bus `%s`", rule_name, event_bus_name)
                return
            if rule.targets:
                LOG.debug(
                    "Notifying %s targets in response to triggered Events rule %s",
                    len(rule.targets),
                    rule_name,
                )
            for target in rule.targets:
                arn = target.get("Arn")
                
                event_str = target.get("Input") or "{}"
                event = json.loads(event_str)
                attr = pick_attributes(target, ["$.SqsParameters", "$.KinesisParameters"])
                try:
                    send_event_to_target(
                        arn,
                        event,
                        target_attributes=attr,
                        role=target.get("RoleArn"),
                        target=target,
                        source_arn=rule.arn,
                        source_service=ServicePrincipal.events,
                    )