API Gateway limits log events to 1024 bytes and cannot be increased. The log events larger than 1024 bytes, such as request and response bodies, will be truncated by API Gateway before submission to CloudWatch Logs.
A workaround can be using Lambda proxy integration with API Gateway.
With Lambda proxy integration, API Gateway passes the request as is to the integrated Lambda function with only exception that the order of the request parameters is not preserved.
This request data includes the request headers, query string parameters, URL path variables, payload, body and request context. As Lambda does not truncate log entry, all headers and query string parameters are logged in Lambda's CloudWatch log and can be seen.
Downside of this approach is that lambda would add to the cost.
Read about API Gateway CloudWatch logs here : https://cloudnamaste.com/api-gateway-cloudwatch-logs/