The timeout configuration is a safety measure of every system, which allows it to recover from operations which take too long.
If you increase the overall timeout, the loosen this safety net a bit.
In your case you have 2 timeouts to consider:
* The timeout of jetty (the request from user to AEM)
* the timeout of your external request (from AEM to the external system)
You have to increase the timeout of both.
You should either send data during processing or rather implement this as workflow or sling job. Then you can inititate the job, and regularly check the status. No need to worry about timeouts anymore.