### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

I Created an REST APIGatway with lambda handler using AWS CDK. when I hit the API the response header content type is always  set to "text/html"  instead of `application/json` . even if I changed the response template or added the header to lambda returned response. when I deploy the same stack on AWS account  the response header is fine and work as expected

### Expected Behavior

Response Header content-type to be `application/json`

### How are you starting LocalStack?

  `localstack start` command

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command`)

    `localstack start` command

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

I'm using AWS CDK typescript application and deploy it using `cdklocal` command on localstack

Using CDK follow the steps:
- create  REST ApiGateway
- create Lambda function 
- integrate lambda with APIGatway

On Command line:
- deploy using `cdklocal deploy` command

Finally
- test the endpoint and check the response header content-type




### Environment

```markdown
- OS: Windows 
- LocalStack: latest
```


### Anything else?

When I deploy the same stack on AWS account everything works fine and the response header content-type is `application/json`
LambdaProxyIntegration,
    MockIntegration,
    RequestTemplates,
    ResponseTemplates,
    SnsIntegration,
    VtlTemplate,
)
from localstack.services.awslambda import lambda_api
from localstack.services.kinesis import kinesis_listener
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.aws_responses import (
    LambdaResponse,
    flask_to_requests_response,
    request_response_stream,
    requests_response,
)
from localstack.utils.common import camel_to_snake_case, json_safe
from localstack.utils.http import add_query_params_to_url
LOG = logging.getLogger(__name__)
TARGET_REGEX_PATH_S3_URI = (
    r"^arn:aws:apigateway:[a-zA-Z0-9\-]+:s3:path/(?P<bucket>[^/]+)/(?P<object>.+)$"
)
TARGET_REGEX_ACTION_S3_URI = r"^arn:aws:apigateway:[a-zA-Z0-9\-]+:s3:action/(?:GetObject&Bucket\=(?P<bucket>[^&]+)&Key\=(?P<object>.+))$"
class AuthorizationError(Exception):
    pass
class RequestValidator:
    __slots__ = ["context", "apigateway_client"]
    def __init__(self, context: ApiInvocationContext, apigateway_client):