Yes, I looked at the wrong 2560. So the module should work fine for you, as you say it does when used alone.
I haven't seen the previous discussions, but it would seem that the three output lines from the processor, MOSI, CLK and CS - should just be connected directly to the corresponding Ethernet and SD module pins, but with each slave having its own CS. However the MISO outputs of Ethernet and SD are both driving that input at the processor, and the question arises as to the state of a MISO output if its CS is turned off. Is it tri-state or open-drain as it should be, or is it an active output high or low? If either MISO output is active even when its CS is off, that could be the problem. And actually, that could even vary with the brand of SD card used since some may have open-drain outputs on MISO while others may not.
If a MISO is active high when its CS is off, you could insert a diode into that line so that it can bring the line low when it needs to, but cannot force the line high and interfere with the other MISO. The SD module already has a 10K pullup resistor on MISO, so you wouldn't need to add that.
If this idea has already been covered before, then I don't know what to suggest.