Definitely stick with a separate 12V supply; Six 1.5V batteries would start at around 9V, but steadily drop down to roughly 6V over their lifetime, causing the LEDs to progressively dim.
This chart is for a typical AA cell, triple A would have a rather shorter life at a given load..
So each group of three LEDs would consume only 20mA and require 1 current limit resistor. You might even be able to go back to using the 2N3907, depends on the gain of the transistor.
Are you not going to use 8 Leds for each?
You can put the last two of each in their own group with their own resistor.
Also be aware the TIP122 has a lot of gain, so with it you can use a larger value base resistor, which should make the capacitor size smaller. You should test these different configurations before settling on specific values.
The only thing I can tell you is that the TIP122 will deliver 20mA to each led without making the base resistor so small (and the cap so large) that you loose the fade on/off effect. You'll have to experiment with what works for you.
You might look at using mosfets instead of the 2N3907s. The gate (the base pin of a mosfet) current is effectively zero, so you could increase the timing resistor to 100K (or higher,) with a similarly reduced capacitance.