### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

When creating an API Gateway with an integration of type AWS to S3, the uri to get objects from an S3 bucket is not working, returning:

`{
    "status": 400,
    "body": "{\"Type\": \"User\", \"message\": \"Error invoking integration for API Gateway ID '3en931aay2': API Gateway AWS integration action URI \\\"arn:aws:apigateway:us-east-1:s3:action/GetObject&Bucket=saxpress-drawio=Key={item}\\\", method \\\"GET\\\" not yet implemented\", \"__type\": \"InvalidRequest\"}",
    "headers": {
        "x-amzn-errortype": "InvalidRequest",
        "Content-Type": "text/html; charset=utf-8",
        "Content-Length": "280"
    }
}`

### Expected Behavior

Retrieve S3 object.

### 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`)

```
localstack:
    image: localstack/localstack:latest
    hostname: localstack
    environment:
      - AWS_DEFAULT_REGION=us-east-1
      - EDGE_PORT=4566
      - HOSTNAME_EXTERNAL=localstack
      - SERICES=s3,lambda,apigateway,apigatewayv2
      - DEBUG=1
    ports:
      - '4566-4583:4566-4583'
      - '8055:8080'
    volumes:
      - 'saxpress-localstack-data:/tmp/localstack'
      - '/var/run/docker.sock:/var/run/docker.sock'
      - './.localstack:/docker-entrypoint-initaws.d'
```

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

```
awslocal apigateway create-rest-api --region us-east-1 --name ${API_NAME} --endpoint-url=http://localhost:4566

awslocal apigateway create-resource \
  --region ${REGION} \
  --rest-api-id ${API_ID} \
  --parent-id ${PARENT_RESOURCE_ID} \
  --path-part "{proxy+}"

awslocal apigateway put-method \
  --region ${REGION} \
  --resource-id ${RESOURCE_ID} \
  --rest-api-id ${API_ID} \
  --http-method "GET" \
  --authorization-type "NONE" \
  --request-parameters "method.request.path.proxy=true"

awslocal apigateway put-integration \
  --region ${REGION} \
  --resource-id ${RESOURCE_ID} \
  --rest-api-id ${API_ID} \
  --type "AWS" \
  --http-method "GET" \
  --uri "arn:aws:apigateway:${REGION}:s3:action/GetObject&Bucket=${BUCKET_NAME}=Key={item}" \
  --credentials "${ROLE_ARN}" \
  --integration-http-method "GET" \
  --request-parameters "integration.request.path.item=method.request.queryString.proxy,integration.request.path.object=context.requestId"

awslocal apigateway put-method-response \
  --region ${REGION} \
  --resource-id ${RESOURCE_ID} \
  --rest-api-id ${API_ID} \
  --status-code "200" \
  --http-method "GET" \
  --response-parameters "method.response.header.Content-Type=true" \
  --response-models "application/json=Empty"

awslocal apigateway put-integration-response \
  --region ${REGION} \
  --resource-id ${RESOURCE_ID} \
  --rest-api-id ${API_ID} \
  --status-code "200" \
  --http-method "GET" \
  --response-parameters "method.response.header.Content-Type=integration.response.header.Content-Type"
```

Testing the method:
```
root@localstack:/opt/code/localstack# awslocal apigateway test-invoke-method --rest-api-id ${API_ID} --resource-id ${RESOURCE_ID} --http-method GET --path-with-query-string index.html
{
    "status": 400,
    "body": "{\"Type\": \"User\", \"message\": \"Error invoking integration for API Gateway ID '3en931aay2': API Gateway AWS integration action URI \\\"arn:aws:apigateway:us-east-1:s3:action/GetObject&Bucket=saxpress-drawio=Key={item}\\\", method \\\"GET\\\" not yet implemented\", \"__type\": \"InvalidRequest\"}",
    "headers": {
        "x-amzn-errortype": "InvalidRequest",
        "Content-Type": "text/html; charset=utf-8",
        "Content-Length": "280"
    }
}
```

### Environment

```markdown
- OS: macOS Big Sur
- LocalStack: latest
```


### Anything else?

_No response_
if not stage:
        return {}
    region_name = [name for name, region in apigateway_backends.items() if api_id in region.apis][0]
    api_gateway_client = aws_stack.connect_to_service("apigateway", region_name=region_name)
    try:
        response = api_gateway_client.get_stage(restApiId=api_id, stageName=stage)
        return response.get("variables")
    except Exception:
        LOG.info(f"Failed to get stage {stage} for api id {api_id}")
        return {}
def get_event_request_context(invocation_context: ApiInvocationContext):
    method = invocation_context.method
    path = invocation_context.path
    headers = invocation_context.headers
    integration_uri = invocation_context.integration_uri
    resource_path = invocation_context.resource_path
    resource_id = invocation_context.resource_id
    set_api_id_stage_invocation_path(invocation_context)
    relative_path, query_string_params = extract_query_string_params(
        path=invocation_context.path_with_query_string
    )
    api_id = invocation_context.api_id
    stage = invocation_context.stage
    source_ip = headers.get("X-Forwarded-For", ",").split(",")[-2].strip()
    integration_uri = integration_uri or ""
    account_id = integration_uri.split(":lambda:path")[-1].split(":function:")[0].split(":")[-1]
    account_id = account_id or TEST_AWS_ACCOUNT_ID
    domain_name = f"{api_id}.execute-api.{LOCALHOST_HOSTNAME}"