Property testing of triangular properties
Your problem is well-understood in the case $S = \{0,1\}$.  Here, the function $\phi$ can be viewed as the adjacency matrix of a graph $G_\phi$ on the vertex set $V$.  And $\phi$ satisfying a triangular property $P$ just means that $G_\phi$ is free of some collection $C_P$ of induced subgraphs, each on $3$ vertices.  Now, it is known [Alon-Fischer-Krivelevich-Szegedy '99] that any such property is testable (in the property testing sense) with query complexity, independent of $n$, bounded from above by a tower of exponentials of height polynomial in $1/\epsilon$.  In particular, when $P$ is 1 only when its input is $(1,1,1)$, the property corresponds to triangle-freeness in graphs for which the best upper-bound is the tower of exp's I mentioned and the best lower bound is slightly superpolynomial in $1/\epsilon$ [Alon '01].  If you want better dependence on $1/\epsilon$ at the cost of perhaps having some dependence on $n$, I don't believe anything much better than what you described is known.  The test you described is more or less folklore.

For arbitrary fixed $S$, I don't know if the testability question has been resolved.  I believe the framework of graph limits, especially the paper [Borgs-Chayes-Lovász-Sos-Szegedy-Vesztergombi '06], might show testability for hereditary properties of weighted graphs which would imply testability of your triangle properties.  But I don't know this fact for sure.