Increasing the voltage by that much will significantly increase the LED current - much more than you might think.
Most of those strips have 3- LEDs in series per section. The forward voltage on a Green or Blue LED is about 3.2V. The Red LEDs are about 1.7V each.
The intended voltage for those strips is about 12V. Let's use that as a starting point.
The increase in current for the Green and Blue LEDs is going to be about:
@ 12V: 12V - (3 * 3.2V) = 2.4V across the current limit resistors
@ 14V: 14V - (3 * 3.2V) = 4.4V across the current limit resistors
@ 16V: 16V - (3 * 3.2V) = 6.4V across the current limit resistors
So: increasing the supply voltage from 12V to 14V is going to increase the LED current by (4.4 / 2.4 * 100%) = 183%
Increasing the supply voltage from 12V to 16V is going to increase the LED current by (6.4 / 2.4 * 100%) = 267%
The situation is a little better with the Red LEDs. Doing the same math as above results in the following voltages across the current limit resistors for the Red LEDs:
12V: 6.9V across the current limit resistor
14V: 8.9V across the resistor
16V: 10.9V across the resistor.
That results in a 129% increase in current if running at 14V; a 158% increase in current if running at 16V.
I strongly suspect that the current limit resistors are going to be really unhappy. You will most likely also notice that the Red LED didn't get as bright as the Green & Blue LEDs at the higher voltages.
Bottom line: run the strips at the manufacturer's maximum voltage and feed power from both ends if possible.
At full white you are looking at 30 amps 12V. I doubt your supply handles 30 amps on the 12v rail alone. And that would fry your controller.
you need a supply that can provide 400W or so or more on the 12v alone.
Look into the rgb power amplifiers or power injectors. These take the signal from the controller and provide separate control for another strip.
Your text does not mention the power amplifiers. But the diagram does. As long as your supply can safely do 360 watts, you should be good. If anything, you might want to power the strips from the middle instead of the end, to ensure better uniformity in color. But you have it handled.
Best Answer
You want to connect it in parallel and you don't need any additional components.
To compute the value you need you just have to use the formula C = (i * dt)/dV, where dV is the value of voltage variation you want and i the current that will be flowing in your leds. From that you should get an approximate value you can try. Try to get the biggest possible value, but be aware that the bigger a capacitor, the bigger the delays.