Can cysteine alone change pigmentation?
Surprisingly, it is indeed possible! But, the fact is that the actual process is a bit more complex and might actually require more compounds along with cysteine to give such effects in living beings. First of all, see this image1:



As is clear from the diagram, GSH (glutathione) is also a factor for pheomelanin production, along with cysteine. Another factor on which one might not pay heed is Tyrosinase protein expression. High amount of L-cysteine is thought to decrease the expression of tyrosinase in cells2 (but since I found just one article, I'd suggest to not pay much attention on this point). 

Apart from gene expression, cysteine affects tyrosinase itself too. High concentrations of cysteine and glutathione have been shown to inactivate tyrosinase in cells, and the cause has been suggested to be the thiol-group of GSH and cysteine, which somehow generates oxygen radicals and inactivates tyrosinase3. A concentration of 10mM cysteine is sufficient to inhibit tyrosinase activity in cells4. It is because of these kinds of effects that GSH and cysteine are now being used in cosmetics as skin whitening agents as they, in higher concentrations, inhibit tyrosinase activity and hence, melanin production in skin cells5.

Just to add, when it comes to gene regulation, then it seems that cysteine has some more effects too. As pointed out by @Chris, eumelanin production in skin is stimulated by a cascade which involves a receptor known as melanocortin-1 receptor (or MC1R). When activated by one of the variants of MSH, typically $\alpha$-MSH, MC1R initiates a complex signaling cascade that leads to the production of the brown or black pigment eumelanin6. Now, a research has concluded that 0.07 mg/ml concentration of L-cysteine greatly reduces expression of MC1R gene, thus literally leaving no way (not really) for eumelanin production7!

Thus, in short, it is not as simple as "Hey, you want a red skin tone? Here is some cysteine for you!" Cysteine, and other related compounds like GSH, have more complex effect on melanin production than just promoting production of pheomelanin over eumelanin. You can also have a look at this answer (by @Chris) for some more information.

EDIT: To tell where you're going wrong, I'll make two reaction flowcharts. First, this is (IMO) what you're expecting as an answer:



But this is what is actually happening:



The red line is what we're interested in. But as you can see, the diagram is already quite complicated (and expected to get more complicated as more factors are discovered). So, giving a simpler explanation than this would be, very much, impractical or impossible.

References:


Malathi M, Thappa DM. Systemic skin whitening/lightening agents: What is the evidence?. Indian J Dermatol Venereol Leprol 2013;79:842-6
The Inhibition Effect of L-cysteine on Melanogenesis in B16F10 Mouse Melanoma Cells; In-Sook An, Ji-Hye Kim, Hoe-sook Yoo, Rui Zhang, Sang-Mo Kang, Tae-Boo Choe, Tae-Jong Kwon, Sung-kwan An, Gi-Yeon Kim
The effect of catalase on the inactivation of tyrosinase by ascorbic acid and by cysteine or glutathione; Lindbladh C, Rorsman H, Rosengren E.
Inactivation of human tyrosinase by cysteine. Protection by dopa and tyrosine.
Jergil B, Lindbladh C, Rorsman H, Rosengren E.
Mechanisms Regulating Skin Pigmentation: The Rise and Fall of Complexion Coloration; Jody P. Ebanks, R. Randall Wickett, and Raymond E. Boissy
Melanocortin 1 receptor - Wikipedia
Substantial Effect of Melanin Influencing Factors on in vitro Melanogenesis in Muzzle Melanocytes of Differently Colored Hanwoo; Touseef Amna, Kyoung Mi Park, In-Kyung Cho, Tae Jeong Choi, Seung Soo Lee, Kang-Seok Seo, and Inho Hwang