I have the same NIC (Realtek 8168) that generally works ok with r8169 driver (Asus motherboard)
but it would sometimes be slow to connect after boot, so i changed to kernel driver r8168
I know Linux users with the same NIC on Gigabyte motherboards where r8169 driver does not work at all and r8169 driver must be blacklisted to use the r8168 driver
suggest try changing to r8168 kernel driver in
manjaro-settings-manager -m msm_mhwd
enable option in bottom left corner - Show All Devices
Look in section Network controller > RTL8111/8168/8411 for network-r8168
Right click on network-r8168 and select Install
Manjaro Settings Manager should install r8168 driver for each kernel version on system and blacklist the r8169 driver