I made the necessary changes base on https://aur.archlinux.org/packages/r8125/

$ sudo pacman -Syyu
$ sudo pacman -Sy base-devel
$ sudo pacman -S $(pacman -Qsq "^linux" | grep "^linux[0-9]*[-rt]*$" | awk '{print $1"-headers"}' ORS=' ')
$ reboot

$ git clone https://github.com/kac89/r8125.git
$ cd r8125
$ makepkg
$ ls
$ sudo pacman -U r8125-dkms-9.003.05-0-x86_64.pkg.tar.gz

(for the -U parameter use the r8125-dkms-X.XXX.XX-X-x86_64.pkg.tar.gz filename that the ls command has displayed, it can be different)

now blacklist the r8169 module (it wil not be loaded anymore)

$ sudo bash -c 'echo "blacklist r8169" > /etc/modprobe.d/r8169.conf'

reboot now, the module r8125 should be loaded automatically. to check it :

$ lsmod | grep r8125

it should return something like : r8125 176128 0

if the module was not loaded automatically, you can force it to be loaded on boot:

$ sudo bash -c 'echo "r8125" > /etc/modules-load.d/r8125.cfg'

now reboot and check if the module is loaded.