### 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_
RequestParametersResolver,
    extract_path_params,
    extract_query_string_params,
    get_event_request_context,
    make_error_response,
    multi_value_dict_for_list,
)
from localstack.services.apigateway.templates import (
    MappingTemplates,
    RequestTemplates,
    ResponseTemplates,
)
from localstack.services.stepfunctions.stepfunctions_utils import await_sfn_execution_result
from localstack.utils import common
from localstack.utils.aws import aws_stack
from localstack.utils.aws.arns import extract_region_from_arn
from localstack.utils.aws.aws_responses import (
    LambdaResponse,
    request_response_stream,
    requests_response,
)
from localstack.utils.aws.client_types import ServicePrincipal
from localstack.utils.aws.templating import VtlTemplate
from localstack.utils.collections import dict_multi_values, remove_attributes
from localstack.utils.common import make_http_request, to_str
from localstack.utils.http import add_query_params_to_url, canonicalize_headers, parse_request_data
from localstack.utils.json import json_safe
from localstack.utils.strings import camel_to_snake_case, to_bytes
LOG = logging.getLogger(__name__)
class IntegrationAccessError(ApiGatewayIntegrationError):