I don’t understand why you would need to mess with the MOSI line. It’s a Master SPI output that drives the relevant slave SD and RFID inputs at the same time, but only the one with its CS low should be paying attention. So I don’t see the need for a resistor.
But the MISO pin is another matter. Both the SD and the RFID modules drive this line. But it should be an active output only from the one with its CS active (low). However, your microSD module likely has a voltage translator chip, and if the SD MISO line goes through a gate of this chip, the output may be driven high by the translator even if the SD itself is open drain or tristate because its CS line is off.
Something similar could be taking place with the RFID module, in which case you may have the two MISO outputs being actively driven in opposite directions. Bus contention they call it. If both modules work correctly when the other is absent, but fail when both are present, this seems to me to be the most likely explanation.
One solution might be to install a pullup resistor on the processor’s MISO pin, then connect that pin to the SD and RFID pins through diodes - oriented so that either module can bring the line low, but neither can force it high. That should eliminate any contention. I’ll attach a schematic showing changes for the SD module.
And you may also need a pullup resistor on the SD card’s MISO pin. This would prevent the input of the voltage translator gate from floating if the SD’s MISO pin floats when its CS is off. Some SD cards behave that way, others do not, but to be safe, you would want to provide for this possibility so all brands of SD cards would work.
Edit: Another possibility if you can physically get to the pin is to disconnect the output enable pin for the relevant gate of the level shifter chip from ground (always enabled), and connect it instead to the CS pin. Then you wouldn’t need any diodes or resistors, and the shifter output would be tristate unless CS is active, which is exactly what we want. Actually, these modules should have been designed that way in the first place.