multiplexing would be in a frequency about 100-200 times/sec (enough to fool the eye that both displays are constantly on), any transistor can do that and even 1000 times that and much more.
You can use a mosfet but there is no need for that, this is usually done with one transistors for each display in common cathode but you have common anode and you may need two so that you can turn off the transistor that drives the display.
Any transistor will do the job as long as it can work with the needed current which should probably be about 100mA when all segments are on.
Alex