How to define a FHE scheme whose plaintext space is infinite using boolean circuits?
I believe we have a more fundamental problem with we try to handle arbitrary length plaintexts.

Suppose that we have any encryption scheme $\mathbb{B}$ (not necessarily FHE, or even asymmetric) that tries to handle arbitrary length plaintexts.  Suppose also we also had a $\mathbb{B}$ encrypted ciphertext, which we can see is 1000 bits long.  We don't know the key, however, we can deduce that there are, at most $2^{1000}$ possible plaintexts that this might correspond to.

What does that deduction give us (given that we don't know the value of any of those $2^{1000}$ plaintexts?  Well, it'd be a pretty good guess that the plaintext isn't more than 1 Gigabyte long, even if the decryption process could potentially have a plaintext which is longer than the ciphertext.

Because of this, there is a limit to the amount of length hiding we can do, if we try to handle unbounded plaintexts.