WWW::Mechanize is up to version 1.34 ... (if you really can't upgrade for whatever reason, you can try to subclass it yourself to add update() taken from the source of a newer version)

my $mech = WWW::Mechanize->new();
$mech->get('.');  # to make it internall create request/response objec
+ts
$mech->update_html($html);
# then use normally
[download]

WWW::Mechanize::Cached may be of interest as well