Farmer's will have no effect on the stain.
Farmer's when made up of a mixture of Ferri & Hypo is a 'cutting' reducer - removing the same amount of silver from highlights and shadows and rapidly causing loss of shadow detail. When used on prints it will clear highlights or fog without effecting midtones.
Farmer's when used as a two step process, first Ferri and then Hypo, is a (semi) 'proportional' reducer - removing the same proportion of silver from highlights and shadows. A proportional reducer can also be made from permanganate and persulfate, the action can be tuned to your needs by varying the proportion of the two ingredients.
Am. Persulfate by itself is a super-proportional reducer that will attack a negative's highlights while leaving the shadows, relatively, alone.
Google for more information - most of the information on reducers is in very old photographic texts. Reducers and intensifiers were much more popular in the days before we had light meters, modern emulsions and reliable developers.
Practice first on scrap negatives. It is a good idea to create some negatives identical to the ones you want to reduce for use as practice. Determine the time/temperature that gives the results you want, don't reduce the real negatives by sight.
The pyro stain is permanent. It can be stopped from forming by the presence of sulfite, and it is intensified by an alkaline bath after fixing. The phenols formed by staining developers polymerize with the gelatin protein - it takes a while for the polymerization process to take place but by the time the film is dry after processing the deed is done.
The famous 'pyro fog' is entirely stain. It is formed when the developer as a whole oxides in air and when the developer oxidizes it forms the above mentioned poly-phenols. When it forms image stain it is because the developer is oxidized in the action of reducing the silver salts to metallic silver, and thus the amount of stain produced is in direct proportion to the amount of silver that is developed.