The image I provided already includes the Realtek R8169 driver mentioned in the Waveshare docs and according to your log the ethernet adapter was detected just fine as eth1:
Code
Sep 03 18:37:19.569160 LibreELEC kernel: r8169 0001:04:00.0: enabling device (0000 -> 0002)
Sep 03 18:37:19.577761 LibreELEC kernel: r8169 0001:04:00.0: can't read MAC address, setting random one
Sep 03 18:37:19.588967 LibreELEC kernel: videodev: Linux video capture interface: v2.00
Sep 03 18:37:19.594613 LibreELEC kernel: r8169 0001:04:00.0 eth1: RTL8168h/8111h, ae:7f:66:a5:82:77, XID 541, IRQ 194
Sep 03 18:37:19.594825 LibreELEC kernel: r8169 0001:04:00.0 eth1: jumbo features [frames: 9194 bytes, tx checksumming: ko]
So you should have an eth1 interface available (check with "ifconfig eth1"), note though that LE doesn't support multiple network interfaces and you have to configure it manually - you may need to adjust whatever you did to set up you USB ethernet adapter.
so long,
Hias