### 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_
cls.fix_proxy_path_params(path_params)
            event["pathParameters"] = path_params
            event["resource"] = resource_path
            event["requestContext"] = request_context
            event["stageVariables"] = invocation_context.stage_variables
            LOG.debug(
                "Running Lambda function %s from API Gateway invocation: %s %s",
                func_arn,
                invocation_context.method or "GET",
                path,
            )
            asynchronous = invocation_context.headers.get("X-Amz-Invocation-Type") == "'Event'"
            return call_lambda(
                function_arn=func_arn,
                event=to_bytes(json.dumps(event)),
                asynchronous=asynchronous,
                invocation_context=invocation_context,
            )
        except ClientError as e:
            raise IntegrationAccessError() from e
        except Exception as e:
            LOG.warning(
                "Unable to run Lambda function on API Gateway message: %s",
                e,
            )
    def invoke(self, invocation_context: ApiInvocationContext):
        uri = (
            invocation_context.integration.get("uri")
            or invocation_context.integration.get("integrationUri")
            or ""