search for RTL8156B or RTL8157 on amazon.de and any of those should work fine for your purposes. See post #
22 above for more details on how to verify if it is working properly or not. Personally I haven't tested a device with an 8157 chip but I have tested multiple 8156B chip based devices.
BTW -- Can you do the following with your existing ethernet and share what you see. On your mac click on top left Apple Icon | About this Mac | More Info | System Report | Ethernet and share the Vendor Name, Product Name, Product ID, Driver, Max Link Speed that you have. Thx