Yes, but not using those modules. WWW::Mechanize is designed to act like a web browser, but if I understand your idea correctly, you need a Perl program to interact with the web browser you the user are already using. For this you can use programmatic control modules. For IE this would be Win32::OLE with the 'InternetExplorer' class or perhaps SAMIE. For Firefox there's some equivalent but I haven't looked into it.

My personal workaround is when I'm about to submit a big text block, I copy and paste it in a text editor before hitting the button. But that also is a bit of a pain.