How BGV's modulus switching works after multiplication?

Please read the BGV paper more carefully. The scale operation is defined in Definition 6. You must ensure that the coefficients of the ciphertext are exact multiples of the scaling factor (what you divide by). You can do that by adding multiples of the plaintext modulus (denoted by r in the paper).

Or find the exact multiple of the plaintext modulus for each coefficient c_j as follows: -c_j \cdot InverseMod[t,q_i]\pmod{q_i}

Note that, AFAIK, almost all existing implementations of BGV implement the variant in the GHS paper.