### 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_
assert event["connection_body_param"] == "value"
        assert headers["Connection-Header-Param"] == "value"
        assert query_args["connection_query_param"] == "value"
        
        assert "/target_path" in event_request.path
        assert event["target_value"] == "value"
        assert headers["Target-Header"] == "target_header_value"
        assert query_args["target_query"] == "t_query"
        
        assert headers["Overwritten-Header"] == "original"
        assert query_args["overwritten_query"] == "original"
        
        match auth["type"]:
            case "BASIC":
                user_pass = to_str(base64.b64encode(b"user:pass"))
                assert headers["Authorization"] == f"Basic {user_pass}"
            case "API_KEY":
                assert headers["Api"] == "apikey_secret"
            case "OAUTH_CLIENT_CREDENTIALS":
                assert headers["Authorization"] == bearer
                oauth_request, _ = httpserver.log[0]
                oauth_login = oauth_request.get_json(force=True)
                
                assert oauth_login["client_id"] == "id"
                assert oauth_login["client_secret"] == "password"
                assert oauth_login["oauthbody"] == "value1"
                assert oauth_request.headers["oauthheader"] == "value2"
                assert oauth_request.args["oauthquery"] == "value3"
    def test_create_connection_validations(self, aws_client):