Although it is true that cognitive assessment is less stable at this age than about four or five years later, it's not completely unreliable, and could still be quite informative, as long as you understand that it's a snapshot of this moment in time, including non-cognitive skills, like sitting still, following directions, attending for long periods of time, and performing on-demand.
The WPPSI-IV and the Brigance are completely different kinds of instruments.
The WPPSI is a norm-referenced instrument, designed to compare a child's performance with a standardization group of representative same-age children. The results tell you what your child's relative standing is estimated to be in the general population of four-year-olds. The Brigance, OTOH, while it does have norm-referenced scores available, was really designed to be a criterion-referenced instrument, assessing mastery of specific skills, rather than comparing to a population. Schools usually find it easier to understand, in terms of double promotion, since they can point to specific skills that are or are not present.
(BTW, the WPPSI also no longer has the verbal-performance dichotomy; there are multiple area scores now.)
What role assessment plays in this early entrance decision is pretty much up to the district. I seem to recall that North Carolina requires students applying for early entry to kindergarten to score at or above the 98th %ile on both a standardized cognitive instrument (such as the WPPSI-IV) and a standardized achievement instrument (such as the WJIV). The Brigance may or may not be considered a qualifying instrument.
http://sbepolicy.dpi.state.nc.us/policies/GCS-J-001.asp?pri=01&cat=J&pol=001&acr=GCS
