Hi
I have had a go a re-crafting the code by @tybalt89 and other comments. It works but I don't know why. It also spits out errors.

The errors indicate $1 for the params hash is not initialised on the line:
 s/^\s*static\s+(\w+)=\K.*/$ip_params{$1}/m or warn "failed to change $1";
I can't see where it is, or should be, initialised.

If you think I should be able to work this out for myself, you are probably right but my excuse is that I am suffering the effects of concussion from a bicycle crash that nearly killed me in February. Writing some software is exercise for my brain in an attempt to restore something like normal service. As part of my mental gymnastics, I have also been refurbishing a vintage RF signal generator. The diary writeup is here: https://www.eevblog.com/forum/repair/wavetek-2025a-0-2-2-200mghz-rf-sig-gen-repair/
As with software, I haven't professionally worked on hardware for decades.

The current code looks like this:

#!/usr/bin/perl

##### FUNCTION
# To read ip settings from a machine generated file and
# write those ip settings to the dhcpcd.conf file.
#  This script is designed to be started by a systemd service.path
#  The service monitors the mb_ip.cfg file for change.  When the file 
+is changed
#  the service calls this script.
#  Updated ip settings are applied after reboot.

##### INPUT
#  Takes user updates of IP settings saved in to the file /home/mimir/
+mb_ip.cfg
#  The user setting are range checked by the app before being saved to
+ file.
#  The file mb_ip.cfg will only have ip settings that are valid (but n
+ot necessarily correct).

##### OUTPUT
#  Writes the ip settings to the /etc/dhcpcd.conf file.
#

#  Reference: https://www.perlmonks.org/?node_id=11136353
#
#  Dazz
#  ver 1.3
#  8 Sept 2021

use strict; 
use warnings;
use Path::Tiny;
use Data::Dump;

###### Input file with ip settings
my $ip_filename = '/home/mimir/mb_ip.cfg';  # The ip info entered by t
+he user into the GUI

###### Output dhcp configuration file
# my $dhcpcdfile = '/etc/dhcpcd.conf';
my $dhcpcdfile = 'd.conf';             # TEST

######################################################################
+######################

use constant { 
  FALSE => 0;
  TRUE  => 1;
};

# Load the input parameters ip_params from the input file
my %ip_params;
%ip_params = ( %ip_params, # add new data to defaults, split with "="
  path($ip_filename)->slurp =~ /^(\w+)=(.*?)\s*$/gm );

# Print the contents of ip_params.                       #TEST
use Data::Dump 'dd'; dd 'ip_params', \%ip_params;        #TEST

# Convert 2 lines DNS to 1 line DNS.
$ip_params{domain_name_servers} = $ip_params{domain_name_server_1}." "
+.$ip_params{domain_name_server_2};
delete($ip_params{domain_name_server_1});
delete($ip_params{domain_name_server_2});

# Print to check the contents ip_params with combined DNS line.       
+                #TEST
use Data::Dump 'dd'; dd 'ip_params', \%ip_params;                     
+ #TEST

{ # block scope for locals
local @ARGV = $dhcpcdfile;
local $^I = '.bak'; # make backup, do inplace edit
my $foundinterface = FALSE;
while( <> )
  {
  if( /^\s*profile\s+static_$ip_params{interface}\b.*\n/m ) # look for
+ profile with matching interface name
    {                                                       # format m
+atches 'static profile_eth0'
    $foundinterface = TRUE;
    warn "found section for $ip_params{interface} at line $.\n";
    }
  elsif( $foundinterface and /static/ )
    {
warn "ip param key : $ip_params{$0}\n"; #TEST 
warn "ip param val : $ip_params{$1}\n"; #TEST ***ERROR:  $1 not initia
+ted. Lines 72 and 73***
    s/^\s*static\s+(\w+)=\K.*/$ip_params{$1}/m or warn "failed to chan
+ge $1";  # match string before \K  Return string after \K
    # ?? What increments $ip_params through the hash ??
    # ?? What stops over-writing of other later sections ?? eg. wlan0
    warn "set $1 to $ip_params{$1}\n";
    }
  elsif( $foundinterface )
    {
    $foundinterface = FALSE;
    warn "ending changes at line $.\n";
    }
  # Output line to dhcpcdfile
  print;
  }
}
[download]

The input file is this: (slightly different to the previous version)

#IP Configuration
#Mon Sep 06 14:29:34 NZST 2021
routers=192.168.9.91
interface=eth0
domain_name_server_2=8.8.4.9
domain_name_server_1=8.8.8.9
ip_address=192.168.10.9/24
[download]

The abbreviated output file to be updated is this:

#static domain_name_servers=192.168.1.1

# fallback to static profile on eth0
#interface eth0
#fallback static_eth0
 
### Static LAN IP profile for Mimirbox
profile static_eth0
 static ip_address=1.1.1.1/24
 static routers=1.1.1.1
 static domain_name_servers=1.1.1.1 2.2.2.2

# Access Point Static interface mimirbox wlan0
  interface wlan0
  static ip_address=192.130.2.20/24
[download]

If these 3 files are placed in the same directory, the code should run.