Function of 'organic' vs regular pipe cleaner
Congratulations on being environmentally sensitive! 

The problem with sink drains is the device that lifts and lowers the stopper. It is a rod that protrudes into the drainpipe above the U-trap; it catches hair and provides a bacterial nest and eventually slows water drainage.

Older drain cleaners were based on alkali to dissolve hair; some used aluminum to foam and break up the hairy mass. Bleach plus alkali works well too, and kills bacteria. The newer, less aggressive formulas can use thioglycolate to dissolve hair (also used on human legs!). 

In the overall scheme of life, with all the other bodily wastes going down the drain, a little drain cleaner may not be a big problem, and where it is important, the government usually has a statement or a regulation. 

Even without regulations tho, it would be kind to the environment to limit our use of toxic chemicals. Using a lot of ineffective product may be worse than using a little of a more aggressive product. I would suggest two approaches: 

1) Pour just a little of the effective product down the drain, but only when it drains slowly (you can measure the drain speed by filling the bowl to a certain line, then timing how long it takes to empty). This little bit (10-20 mL?) will saturate the clog because it is higher than the bottom of the U-trap. Give it some time (many minutes, even hours); see if it has opened up. If not, repeat. This may go against product recommendations to use a lot of product, but I suspect that the bottom of the U-trap is not blocked (any water flow usually cleans that out). This will reduce your environmental impact by using your brain.

2) Use a mechanical method. a) A stiff copper wire with a tiny hook at the end is useful for digging out hair, and often opens up the drain with just a lot of hot water. b) Cut a hollow rubber ball in half to make a tiny plunger. Place it over the drain (with the stopper lifted a little) and press it up and down to push the clog all the way down the drain.