### 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_
expected_fields = (
            "version",
            "id",
            "detail-type",
            "source",
            "account",
            "time",
            "region",
            "resources",
            "detail",
        )
        for field in expected_fields:
            assert field in event
    def test_put_rule(self, aws_client):
        rule_name = f"rule-{short_uid()}"
        aws_client.events.put_rule(Name=rule_name, EventPattern=json.dumps(TEST_EVENT_PATTERN))
        rules = aws_client.events.list_rules(NamePrefix=rule_name)["Rules"]
        assert len(rules) == 1
        assert json.loads(rules[0]["EventPattern"]) == TEST_EVENT_PATTERN
        
        self.cleanup(rule_name=rule_name)
    def test_events_written_to_disk_are_timestamp_prefixed_for_chronological_ordering(
        self, aws_client