### 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`
if response and response.content is not None:
            response.headers["Content-Length"] = str(len(response.content))
    def invoke(self, invocation_context: ApiInvocationContext):
        uri = (
            invocation_context.integration.get("uri")
            or invocation_context.integration.get("integrationUri")
            or ""
        )
        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
        headers = invocation_context.headers
        resource_path = invocation_context.resource_path
        invocation_context.context = get_event_request_context(invocation_context)
        try:
            path_params = extract_path_params(path=relative_path, extracted_path=resource_path)
            invocation_context.path_params = path_params
        except Exception:
            path_params = {}
        func_arn = uri
        if ":lambda:path" in uri:
            func_arn = uri.split(":lambda:path")[1].split("functions/")[1].split("/invocations")[0]
        if invocation_context.authorizer_type:
            authorizer_context = {
                invocation_context.authorizer_type: invocation_context.auth_context