In order to generate my BSK from my LWE keys of parameters LWE128_630, you use RLWE128_1024_1 in the examples. I understand that technically a RLWE128_256_1 could encrypt the key also, but is the reason you choose 1024 because it has a smaller standard deviation only?
The first step of a (programmable) bootstrapping consists in computing a modulus switching, reducing each coefficient of your LWE ciphertext (a_1,…,a_n, b)
from values up to q=2^64 (or q=2^32) to values up to 2N, where N is the degree of the RLWE. Then, if you choose N=256, this means that after the modulus switch the input ciphertext will encrypt a plaintext of log2(2*256) = 9 bits (including message and noise). In this example, a part of the message will be overwritten with the noise because of picking such a small N.
In practice, starting from N=1024 is generally recommended to ensure that small messages are not overwritten by noise (with high probability).