### 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_
{
        "type": "API_KEY",
        "key": "ApiKeyAuthParameters",
        "parameters": {"ApiKeyName": "Api", "ApiKeyValue": "apikey_secret"},
    },
    {
        "type": "OAUTH_CLIENT_CREDENTIALS",
        "key": "OAuthParameters",
        "parameters": {
            "AuthorizationEndpoint": "replace_this",
            "ClientParameters": {"ClientID": "id", "ClientSecret": "password"},
            "HttpMethod": "put",
            "OAuthHttpParameters": {
                "BodyParameters": [{"Key": "oauthbody", "Value": "value1"}],
                "HeaderParameters": [{"Key": "oauthheader", "Value": "value2"}],
                "QueryStringParameters": [{"Key": "oauthquery", "Value": "value3"}],
            },
        },
    },
]
EVENT_BUS_ROLE = {
    "Statement": {
        "Sid": "",
        "Effect": "Allow",
        "Principal": {"Service": "events.amazonaws.com"},
        "Action": "sts:AssumeRole",
    }
}
@pytest.fixture
def events_allow_event_rule_to_sqs_queue(aws_client):