However at some point during the process, the script abruptly dies.

You could try to unset autocheck in the constructor (and then handle errors yourself, of course):

my $mech = WWW::Mechanize->new( autocheck => 0 );
...
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When enabled (which is the default in newer¹ versions of WWW::Mechanize), any errors make the program die.  You seem to have a newer version...

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¹ from the Changes file:

1.49_01     Sat Sep 27 23:50:04 CDT 2008
========================================
[THINGS THAT MAY BREAK YOUR CODE]
The autocheck argument to the constructor is now ON by default,
unless WWW::Mechanize is being subclassed.  There are so many new
programmers whose ->get() calls fail unchecked that I'm now putting
on the seat belts for them.
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