With only a single LED lit, it isn't important that it has an absolutely perfectly controlled intensity. The total power dissipation will also be low.

The simplest way is probably to use 5 8-bit shift registers. Send out the data and latch it. Or make it even simpler - just send out 40 bits without use of output-enable or latch. The processor is fast enough that no one will be able to see the glow when the data is shifted to the correct position.

Since only one LED is expected to be lit, you can connect one side of all diodes to a single resistor.

If the shift registers have an OE signal - or if you have a common NPN transistor to GND - you can use PWM to adjust the intensity of the LED.

Your processor don't need to constantly refresh anything - the shift registers keeps the requires state as long as you don't play with their input signals.