I tried numerous ways but the only one I found that preserved the style was to store it on a first pass and re-apply it on a second pass like this.

my $paragraphs = $file->Paragraphs();
my $n = $paragraphs->Count();
my @style=(); # add
for my $p (1..$n) {
  my $paragraph = $paragraphs->Item( $p );
  print "paragraph $p\n";
  print "before: $paragraph->{Style}->{NameLocal}\n";
  $style[$p] = $paragraph->{Style}->{NameLocal}; # add
  my $text = $paragraph->{Range}->{Text};
  $paragraph->{Range}->{Text} = $text;
}


for my $p (1..$n) {
  my $paragraph = $paragraphs->Item( $p );
  $paragraph->{Style} = $style[$p]; # add
  print "after: $paragraph->{Style}->{NameLocal}\n";
}
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