I have not seen any integrated amplifier that has an Ethernet input except for the Auralic Polaris which does not have XLR inputs and is outside your budget . It is mostly streaming devices that use Ethernet.
Once you move your computer closer to your system why not use the USB output from the computer to the amplifier. Any integrated with a built in DAC will have that option. The Parasound Halo integrated checks off all of your boxes that way, but no Ethernet. They are regularly listed for less than $2K used. Good luck!
Once you move your computer closer to your system why not use the USB output from the computer to the amplifier. Any integrated with a built in DAC will have that option. The Parasound Halo integrated checks off all of your boxes that way, but no Ethernet. They are regularly listed for less than $2K used. Good luck!

