How does TYRP1 affect the synthesis of black and brown eumelanin (on a biochemical level)?
The actual coat color is determined by the ratio of the products of two genes: Tyr and TyrP1. The first oxidizes DHI to Indole-5,6-quinone and forms brown pigment, TyrP1 on the other hand oxidizes DHICA to Indole-5,6-quinone carbolic acid, which forms black pigment. See this pathway diagram (from here): 



So loss of TyrP1 activity is not harmful to general pigmentation but prevents the formation of black color, while the inhibition (or loss of function) for Tyr will lead to albinism, since no pigment precursors (from Tyrosine to DOPA can be formed.

You may also be interested in this paper: "TYRP1 and MC1R genotypes and their effects on coat color in dogs."