### 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_
import base64
import json
import logging
import re
from abc import ABC, abstractmethod
from functools import lru_cache
from http import HTTPStatus
from typing import Any, Dict
from urllib.parse import urljoin
import requests
from botocore.exceptions import ClientError
from moto.apigatewayv2.exceptions import BadRequestException
from requests import Response
from localstack import config
from localstack.aws.accounts import get_aws_account_id
from localstack.aws.connect import (
    INTERNAL_REQUEST_PARAMS_HEADER,
    InternalRequestParameters,
    connect_to,
    dump_dto,
)
from localstack.constants import APPLICATION_JSON, HEADER_CONTENT_TYPE
from localstack.services.apigateway import helpers
from localstack.services.apigateway.context import ApiInvocationContext
from localstack.services.apigateway.helpers import (
    ApiGatewayIntegrationError,
    IntegrationParameters,