### 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_
_stream = aws_client.kinesis.describe_stream(StreamName=stream_name)
            assert _stream["StreamDescription"]["StreamStatus"] == "ACTIVE"
        
        retry(check_stream_status, retries=7, sleep=0.8)
        aws_client.events.put_events(
            Entries=[
                {
                    "EventBusName": bus_name,
                    "Source": TEST_EVENT_PATTERN["source"][0],
                    "DetailType": TEST_EVENT_PATTERN["detail-type"][0],
                    "Detail": json.dumps(EVENT_DETAIL),
                }
            ]
        )
        stream = aws_client.kinesis.describe_stream(StreamName=stream_name)
        shard_id = stream["StreamDescription"]["Shards"][0]["ShardId"]
        shard_iterator = aws_client.kinesis.get_shard_iterator(
            StreamName=stream_name,
            ShardId=shard_id,
            ShardIteratorType="AT_TIMESTAMP",
            Timestamp=datetime(2020, 1, 1),
        )["ShardIterator"]
        record = aws_client.kinesis.get_records(ShardIterator=shard_iterator)["Records"][0]
        partition_key = record["PartitionKey"]
        data = json.loads(record["Data"].decode())
        assert partition_key == TEST_EVENT_PATTERN["detail-type"][0]
        assert data["detail"] == EVENT_DETAIL
        self.assert_valid_event(data)
    def test_put_events_with_input_path(self, aws_client):