Since all evaluation in BGV scheme is on polynomials it seems that it does not really matter if the coefficients of the polynomials are within $q$ range or not. All must be okay for the congruent values.
I have the following questions:
If there are no specific optimizations (e.g. RNS) for BGV, can we work in modulo $Kq$, where $K$ is some arbitrary number? Let us ignore encryption/decryption and focus only on adversary side evaluations: additions/multiplications.
If the answer to question 1 is yes, then what optimizations can be affected or broken by working in $Kq$?
Is it the same for BFV scheme?