Zama says it’s going to support 64-bit exact integers. But for exact computations, we need q=2N or very approximate, so the rescaling is the identity. However, for that to happen for 64-bit numbers, we’d have to have room for lots of errors.

I think Zama plans to do it in small chunks, decomposing the 64-bit number into many ciphertexts. If so, what would be the number of bits for the error and message in each ciphertext?

You are right, at the moment the idea to support large integer messages is to split
them into small chunks. The size of each chunk lies into the range [1 bit, 8 bits].

For instance, for a 64-bit integer, many decompositions are possible: 8 chunks of 8-bit messages, 16 chunks of 4-bit messages, …

From the choice of the number of message bits in each chunk, cryptographic parameters are then chosen accordingly. This gives the number of noise bits per ciphertext.