In this case, will 5V not be enough to handle my two 3.3V LEDs in series pairs?
This was my suggestion: give it a try.
I guess you'll see a voltage drop of 2.5 V at each led and a current of less than 20 mA. No resistor required, probably.
Now the question is if the brightness is still ok.
My suggestion with the two leds in series implies they are controlled by one pin only, thus you just need 7 pins.
If the leds are wired opposite each other on your circle, you'd have two lights ( or black holes or light trains ) chasing each other around.
If the leds are next to each other, you'd just have an effective granularity of 7 elements only, but can build one "light train" running around.
(Initially, I imagined the first suggestion, or even fewer bigger groups up to 3groups with 5 leds, giving 5 lights chasing each other faster and faster until they seem to shine continuously at or above 20 Hz, when you switch to PWM output pulsing the whole circle)