Update. PDF::API2 supports incremental PDF modification only, raw file size can only grow, comparing "snapshots" is useless. Modified objects must be marked as such, so they'll be "slapped" after original %%EOF on save. That, mostly, is what patch does (plus a few more things), all "setter" methods should include $self->{' api'}->{'pdf'}->out_obj($self);, and/or similar line if parent/siblings are modified. I stopped patching half way, now I see your particular intent was to change bookmark titles, then patch title setter too, it'll work then. Note PDF::API2::Outline::outline always adds a new descendant to invocant; PDF::API2::outline is different.

Line added to "main" module API2.pm is lazy way to ensure objects are re-blessed, so that e.g. $pdf->outline->first->delete won't boom. Two lines commented-out in patch look harmless but useless to me: parent method never called as setter, deleting ' children' array achieves nothing. + See modified PDF content in OP, object number is skipped, it's just not tidy, I wonder if it always was this way. These latter comments are for maintainer (Steve?) if they are interested.

__________

(PDF::API2... patch and fix and patch again). Below there are patches for 2 files in distribution, and your modified test program which produces four PDF files with expected content I think. I'll add later to this node as to why your tests as designed make no sense and why it's only "demo" patch, I gave up patching further because out of time, only your immediate test files are fixed.

--- C:\berrybrew\strawberry-perl-5.32.1.1-64bit-PDL\perl\site\lib\PDF\
+API2.pm.backup    Wed Dec  8 07:53:45 2021
+++ C:\berrybrew\strawberry-perl-5.32.1.1-64bit-PDL\perl\site\lib\PDF\
+API2.pm   Sat Sep  3 13:36:58 2022
@@ -824,6 +824,7 @@
         bless $obj, 'PDF::API2::Outlines';
         $obj->{' api'} = $self;
         weaken $obj->{' api'};
+        $obj->count();
     }
     else {
         $obj = PDF::API2::Outlines->new($self);
         
         
--- C:\berrybrew\strawberry-perl-5.32.1.1-64bit-PDL\perl\site\lib\PDF\
+API2\Outline.pm.backup    Wed Dec  8 07:53:45 2021
+++ C:\berrybrew\strawberry-perl-5.32.1.1-64bit-PDL\perl\site\lib\PDF\
+API2\Outline.pm    Sat Sep  3 16:23:35 2022
@@ -90,25 +90,25 @@
     if ($count) {
         $self->{'Count'} = PDFNum($self->is_open() ? $count : -$count
+);
     }
+    else {
+        delete $self->{'Count'}
+    }
 
     return $count;
 }
 
 sub _load_children {
     my $self = shift();
+    $self->{' children'} = [];
     my $item = $self->{'First'};
-    return unless $item;
-    $item->realise();
-    bless $item, __PACKAGE__;
-
-    push @{$self->{' children'}}, $item;
-    while ($item->next()) {
-        $item = $item->next();
+    
+    while ($item) {
         $item->realise();
         bless $item, __PACKAGE__;
+        $item->{' api'} = $self->{' api'};
         push @{$self->{' children'}}, $item;
+        $item = $item->next()
     }
-    return $self;
 }
 
 =head3 first
@@ -121,8 +121,10 @@
 
 sub first {
     my $self = shift();
-    if (defined $self->{' children'} and defined $self->{' children'}
+->[0]) {
-        $self->{'First'} = $self->{' children'}->[0];
+    if (exists $self->{' children'}) {
+        $self->{'First'} = @{$self->{' children'}}
+            ? $self->{' children'}[0]
+            : undef
     }
     return $self->{'First'};
 }
@@ -137,8 +139,10 @@
 
 sub last {
     my $self = shift();
-    if (defined $self->{' children'} and defined $self->{' children'}
+->[-1]) {
-        $self->{'Last'} = $self->{' children'}->[-1];
+    if (exists $self->{' children'}) {
+        $self->{'Last'} = @{$self->{' children'}}
+            ? $self->{' children'}[-1]
+            : undef
     }
     return $self->{'Last'};
 }
@@ -154,7 +158,7 @@
 
 sub parent {
     my $self = shift();
-    $self->{'Parent'} = shift() if defined $_[0];
+#    $self->{'Parent'} = shift() if defined $_[0];
     return $self->{'Parent'};
 }
 
@@ -167,8 +171,11 @@
 =cut
 
 sub prev {
-    my $self = shift();
-    $self->{'Prev'} = shift() if defined $_[0];
+    my ($self, $other) = @_;
+    if ($other) {
+        $self->{'Prev'} = $other;
+        $self->{' api'}{'pdf'}->out_obj($self);
+    }
     return $self->{'Prev'};
 }
 
@@ -181,8 +188,11 @@
 =cut
 
 sub next {
-    my $self = shift();
-    $self->{'Next'} = shift() if defined $_[0];
+    my ($self, $other) = @_;
+    if ($other) {
+        $self->{'Next'} = $other;
+        $self->{' api'}{'pdf'}->out_obj($self);
+    }
     return $self->{'Next'};
 }
 
@@ -200,7 +210,9 @@
     my $self = shift();
 
     my $child = PDF::API2::Outline->new($self->{' api'}, $self);
-    $self->{' children'} //= [];
+    
+    $self->_load_children() unless exists $self->{' children'};
+    
     $child->prev($self->{' children'}->[-1]) if @{$self->{' children'
+}};
     $self->{' children'}->[-1]->next($child) if @{$self->{' children'
+}};
     push @{$self->{' children'}}, $child;
@@ -208,6 +220,7 @@
         $self->{' api'}->{'pdf'}->new_obj($child);
     }
 
+    $self->{' api'}->{'pdf'}->out_obj($self);
     return $child;
 }
 
@@ -268,6 +281,7 @@
         $item = $item->next();
         push @{$self->{' children'}}, $item;
     }
+    $self->{' api'}->{'pdf'}->out_obj($self);
     return $self;
 }
 
@@ -291,7 +305,8 @@
 
     my $siblings = $self->parent->{' children'};
     @$siblings = grep { $_ ne $self } @$siblings;
-    delete $self->parent->{' children'} unless $self->parent->has_chi
+ldren();
+#    delete $self->parent->{' children'} unless $self->parent->has_ch
+ildren();
+    $self->{' api'}->{'pdf'}->out_obj($self->parent);
 
     return;
 }
[download]

############

#!/usr/bin/perl
use 5.032;
use warnings;
use strict;
use PDF::API2;

my ($stringy_bare_pdf, $stringy_outline_pdf);

BARE_PDF: {
    my $pdf = PDF::API2->new(-compress => 0);
    my $page1 = $pdf->page();
    
    $pdf->save('bare.pdf');
}

OUTLINE: {
    my $pdf = PDF::API2->new(-compress => 0);
    my $page1 = $pdf->page();
    
    my $outlines = $pdf->outlines();
    my $outline = $outlines->outline();
    $outline->title('Test Outline 1');
    $outline->dest(1);
    
    $stringy_outline_pdf = $pdf->to_string();
    
    open my $fh, '>', 'outline.pdf' or die;
    binmode $fh;
    print $fh $stringy_outline_pdf;
    close $fh;
}

DELETE_OUTLINE: { 
    my $pdf = PDF::API2->from_string($stringy_outline_pdf, -compress =
+> 0) or die $!;
    
    my $root = $pdf->outlines();
    my $outline = $root->first();   
    $outline->delete();
    
    $pdf->save('outline_deleted.pdf');
}

MODIFY_OUTLINE: {
    my $pdf = PDF::API2->from_string($stringy_outline_pdf, -compress =
+> 0) or die $!;
    
    my $root = $pdf->outlines();
    my $outline = $root->outline();
    $outline->title('Test Outline 2');
    $outline->dest(1);
    
    $pdf->save('outline_modified.pdf');
}
[download]

Thank you very much for the clarification and the patches

So my code to rename and delete the outlines was correct, but at the end of the day, unpatched PDF::API2 version 2.043 does not currently support renaming or completely deleting existing outlines in a PDF.

Example test relies on incorrect use of PDF::API2::Outline::outline despite its documentation, but code unshown may in fact be correct if bookmark tree is traversed with other (too verbose for example test?) -- i.e. First/Next/Prev/Last -- methods.