Is following SAT case in $\mathsf{P}$?
This answer assumes the following interpretation of your question:


  What is the complexity of the following problem? Decide whether a given CSP is satisfiable, where at most half of the assignments satisfy each constraint.


It is likely that this problem is either in $\mathsf{P}$ or $\mathsf{NP}$-complete. In fact, if we bound the width of the constraints (how many variables appear in them), then Schaefer's dichotomy theorem implies that the problem is either in $\mathsf{P}$ or $\mathsf{NP}$-complete, giving a criterion to decide which of these possibilities holds.

In your case, it seems likely that the problem is $\mathsf{NP}$-complete already for width 3. You can verify (or refute) this using the criterion in Schaefer's theorem, which can be found in the Wikipedia article linked to above.