Grr, that'll teach me to make changes without testing. The substitution needs to have those square brackets escaped:

while (m/\[\[\w+\]\]/) {
    my $oldpos = pos;
    m/\[\[(\w+)\]\]/;
    
    my $expansion = $helper->($1);
    pos = $oldpos;
    s/\[\[$1\]\]/$expansion/;
}
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