How to increase pheomelanin production?
Since Eumelanin is the protective form of melanin and there is some research available that Pheomelanin contributes to the generation of reactive oxygen species which are harmful to the cell (see reference 1, which contains a lot of further references on this topic), I don't see any reason to do so, but ok. Additionally removing Eumelanin from your skin will lead to a higher risk for skin cancers as Melanoma. Let's think through this:

If we look at the biosynthesis pathway of the Melanins, they both start with the same two reactions (see the figure from here): 



The routs parts at the DOPAquinone, which in the presence of Gluthathione or Cysteine goes into the pheomelanin direction. The concentration of Cysteine available to the cell needs to be higher than 0,76 µM (see references 2 and 3 for details), otherwise the reaction goes exclusively into the Eumelanin direction.

There are some skin whitening (or better lightening) products available which contain L-Cysteine or GSH. This will work to some extend, but once the level drops enough, the ratio will shift back to the more natural ratio between Eumelanin and Pheomelanin. There is also no data available on the risks of doing this, so I would strongly advocate against it. See the review in reference 4 for details and additional references on this topic.

If you now think, ok, taking some pills is enough, I would call this a false hope, since this process is highly regulated. This happens on one part through external signaling through the MC1R receptor (which as you already mentioned is of importance here) and also through the availability of key enzymes in the process as well as stability of them and their regulation.

Signaling through the MC1R receptor is highly important for pigmentation and also the amount of enzymes available (the transcription factor MITF is regulated by it). If the signaling is strong, the cysteine storage is soon depleted bringing on the production of Eumelanin. If this signaling is weak 
this will not happen and the production of pheomelanin is favoured. Mutations in MC1R are the reason for people having red hair and and fair skin. See reference 5 and 6. 

Interfering with the signaling cascade which is activated by MC1R is certainly not a good idea either, since this will interfere with other processes in the cell as well. These pathways are also disregulated in cancers, so I would not touch these.

Summarizing I think there will only be a way of influencing this by taking some supplements, but how safe this is, is unknown. All the other possible ways are certainly more dangerous.

References:


How does pheomelanin synthesis contribute to melanomagenesis?: Two
distinct mechanisms could explain the carcinogenicity of pheomelanin
synthesis.
Rate constants for the first two chemical steps of eumelanogenesis.
Chemistry of mixed melanogenesis--pivotal roles of dopaquinone.
Systemic skin whitening/lightening agents: What is the evidence?
UV signaling pathways within the skin
The melanocortin-1 receptor: red hair and beyond.