I do not have a problem with using the SD and w5100 together. Neither does zoomkat. Here is the setup code I use to initialize both correctly.
void setup() {
Serial.begin(9600);
// disable w5100 SPI while starting SD
pinMode(10,OUTPUT);
digitalWrite(10,HIGH);
Serial.print(F("Starting SD..."));
if(SD.begin(4) == 0) Serial.println(F("fail"));
else Serial.println(F("ok"));
Serial.print(F("Starting w5100..."));
Ethernet.begin(mac, ip, gateway, gateway, subnet);
Serial.println(Ethernet.localIP());
Serial.println(F("Ready."));
}
edit: I haven't tried this, but it looks ok to me. This is one of zoomkat's latest examples.