What consequences do the plaintext space size has on the performances in the BGV scheme?
A helpful resource to assess which ring-based FHE scheme (like BGV) would work best for you, see here: https://eprint.iacr.org/2015/889

In this paper they compare four schemes: BGV, FV, NTRU, and YASHE. 

The reason I point you there is that one major factor in the performance of these schemes is the value of p. If a large p is used, BGV outperforms the rest. If a small p is used, YASHE wins, but not by too much. The other main factor is the L parameter. 

The metric they use to compare them is the ciphertext size, which is crude but gives a rough estimate as to the computational time needed.