3SAT to 3SAT-WITH-MAJORITY reduction
Hint: Take a SAT instance $C$ with $k$ clauses and introduce $3k$ new variables $\top^i_1, \top^i_2, \top^i_3$ for $i \leq k$ and construct $k$ new clauses $c^\top_i = \{ \top^i_1,\top^i_1,\top^i_3 \}$.  Let $C' = C \cup \bigcup_{i=1}^k c^\top_i$ be the input to 3SAT-WITH-MAJORITY.