MAKE A CALL TO A REST API WITH QUERYSTRING PARAMETERS

From what I understand, your problem has multiple layers.

Lambda without Proxy Integration

According to the proxy integration documentation, the event object is prepared for you by the proxy integration in the way you said: Set up Lambda proxy integrations in API Gateway.

Without the proxy integration, please consider that the handler gets different content in event object. This would lead to your problem "throws an error that the parameter is undefined".

So for now let's go with Proxy Integration and see about the CORS Problem.

CORS Problem

A GET request is considered a "simple request" in matters of CORS. I found the MDN documentation about CORS very helpful (read from top to bottom once, totally took away the "magic appearance" from CORS): MDN Cross-Origin Resource Sharing (CORS).

Your HTTP response must contain an Allow-Control-Allow-Origin HTTP Header. AWS Documentation should give you more directions: Enabling CORS support for proxy integrations.

export const handler = async (event) => {
    const response = {
        statusCode: 200,
        headers: {
            "Access-Control-Allow-Headers" : "Content-Type",
            "Access-Control-Allow-Origin": "https://www.example.com",
            "Access-Control-Allow-Methods": "OPTIONS,POST,GET"
        },
        body: JSON.stringify('Hello from Lambda!'),
    };
    return response;
};

-- from Enabling CORS support for proxy integrations

I hope this can somewhat shed some light on your situation!