### 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`
uri = integration.get("uri") or integration.get("integrationUri") or ""
    
    
    custom_auth_header = invocation_context.headers.pop(HEADER_LOCALSTACK_AUTHORIZATION, "")
    invocation_context.headers["Authorization"] = custom_auth_header
    try:
        path_params = extract_path_params(path=relative_path, extracted_path=resource_path)
        invocation_context.path_params = path_params
    except Exception:
        path_params = {}
    if (uri.startswith("arn:aws:apigateway:") and ":lambda:path" in uri) or uri.startswith(
        "arn:aws:lambda"
    ):
        if integration_type == "AWS_PROXY":
            return LambdaProxyIntegration().invoke(invocation_context)
        elif integration_type == "AWS":
            func_arn = uri
            if ":lambda:path" in uri:
                func_arn = (
                    uri.split(":lambda:path")[1].split("functions/")[1].split("/invocations")[0]
                )
            headers = helpers.create_invocation_headers(invocation_context)
            invocation_context.context = helpers.get_event_request_context(invocation_context)
            invocation_context.stage_variables = helpers.get_stage_variables(invocation_context)
            if invocation_context.authorizer_type:
                invocation_context.context["authorizer"] = invocation_context.auth_context
            request_templates = RequestTemplates()
            payload = request_templates.render(invocation_context)
            
            result = lambda_api.process_apigateway_invocation(