A forwarder is just a rule for the mailserver, it has neither mail box storage nor credentials to authenticate you for sending mail.
I assume you were thinking of something like this? - using your free Gmail account to send from a different alias (your Cpanel forwarder address) :
Want to get more out of Google apps at work or school?
support.google.com
But this is far from an ideal setup. Deliverability will suffer to an extent, even if you add spf.google.com to your SPF record in the domain's DNS zone. You will find your messages end up in recipients' Junk/Spam folder more often.
And you will not always be able to hide the fact that you are sending from Gmail. E.g. other Gmail users will see messages from you as sent from "
[email protected] via gmail.com".