My @INC is the same as yours.
On my machine the module is here and looks like this
$ ls -l /usr/lib/perl5/vendor_perl/5.10.0/LWP/UserAgent.pm
-rw-r--r--. 1 root root 51803 2009-02-13 15:06 /usr/lib/perl5/vendor_perl/5.10.0/LWP/UserAgent.pm
If it isn't there on your machine you could try installing the appropriate package.
I *think* LWP is part of perl-libwww-perl so try
# yum install perl-libwww-perl
I have a script that uses LWP::UserAgent on my machine and it works just fine.
There is a package in the Fedora repository named "perl-LWP-UserAgent-Determined" I DO NOT have that one installed