A couple options come to mind:

1. There is probably no need to go to the absolute latest version, so look through the archived versions of LWP and see when the proxy auth code was added (the Changes file should help). Perhaps there exists an older version that still works properly with your version of perl.
2. Or install a copy of perl in your homedir and use it instead of the system perl.
3. Or like you mention, install a new version of LWP in a nonstandard dir and throw a 'use lib qw(/path/to/lwpdir/);'. As for the changes that you are likely to need to make, what you mention sounds OK, but you shouldn't need to turn off strict. Just change all 'our $globalvar' vars to 'use vars qw($globalvar);'. 'our' and 'use vars' are not identical, but should be close enough.
4. Or if none of those are workable, then ditch LWP and launch an external program like 'wget' instead. Not ideal or efficient for that matter, but it should work.