Can we proxy-re-encrypt using homomorphic encryption schemes?
YES. It is pointed out by Gentry (in his thesis) when he consider bootstrapping and re-encryption.

Let us denote the old key pair by $(PK, SK)$ and the new key pair by $(PK^1, SK^1)$.

What the user wants is to re-encrypt an old ciphertext $C = \mathsf{Enc}(PK, M)$ into a new ciphertext $C^1$ which is decrypted into $M = \mathsf{Dec}(SK^1,C^1)$.

To do so without direct decryption, we virtually decrypt $C$ in the draft chamber guarded by $PK^1$, that is, we evaluate a decryption circuit with $PK^1$.
Correctly speaking, we do the following:


Generate $\widetilde{SK}_i = \mathsf{Enc}(PK^1,SK_i)$, where $SK_i$ denotes the i-th bit of $SK$.
Compute $\widetilde{C}_i = \mathsf{Enc}(PK^1,C_i)$, where $C_i$ denotes the i-th bit of $C$.
Evaluate the decryption circuit. That is, $C^1 = \mathsf{Eval}(PK^1, \mathsf{Dec}, (\widetilde{SK}_1,\dots,\widetilde{SK}_n,\widetilde{C}_1,\dots,\widetilde{C}_m))$.


From the correctness of evaluation, we have
$$
\begin{align}
\mathsf{Dec}(SK^1,C^1)
 &amp;= \mathsf{Dec}(\mathsf{Dec}(SK^1,\widetilde{SK}_1),\dots,\mathsf{Dec}(SK^1,\widetilde{SK}_n),\mathsf{Dec}(SK^1,\widetilde{C}_1),\dots,\mathsf{Dec}(SK^1,\widetilde{C}_m))\\
 &amp;= \mathsf{Dec}(SK_1,\dots,SK_n,C_1,\dots,C_m) \\
 &amp;= \mathsf{Dec}(SK,C)\\
 &amp;= M
\end{align}
$$
as we want.

This is just re-encryption Gentry suggested in his thesis.