Direkt oberhalb des von mir verlinkten Absatzes gibt es eine Methode "form_with_fields".

perldoc WWW::Mechanize

$mech->form_with_fields( @fields )

Selects a form by passing in a list of field names it must contain. If there is more than one form on the page with that matches, then the first one is used, and a warning is generated.

If it is found, the form is returned as an HTML::Form object and set internally for later used with Mech's form methods such as "field()" and "click()".

Returns undef if no form is found.

Note that this functionality requires libwww-perl 5.69 or higher.

(Hervorhebungen von mir)

Damit sollte es doch möglich sein, das gewünschte Formular auszuwählen, um dann die set_field() anzuwenden...