Compressing BLS signature with a hash function It seems to me, that it should be possible to hash BLS signature to achieve significant space saving. Here is how it could work:

Assuming we have a pairing-friendly elliptic curve with two generator points $G_1$ and $G_2$. Let's say my public key is $P = p \cdot G_1$, where $p$ is my private key. The standard BLS signature of message $m$ would be:

$$
S = p \cdot H(m)
$$

where, $H$ is a hash function that maps the message into the subgroup defined by $G_2$. The verification of the signature is done using a pairing function $e$ as follows:

$$
e(P, H(m)) \stackrel{?}{=} e(G_1, S)
$$

If we use a curve such as BLS12-381, the size of the signature could be 96 bytes. But what if we redefine the signature as:

$$
s = H_2(e(G_1, p \cdot H(m)))
$$

where, $H_2$ is a cryptographic hash function (e.g. SHA256). The verification can then be done as follows:

$$
H_2(e(P, H(m))) \stackrel{?}{=} s
$$

Not only is the signature now only 32 bytes, but it also takes only 1 pairing to verify.

The obvious drawback is that signatures can no longer be aggregated, but I'm wondering if there are any other issues with using this scheme.
but I'm wondering if there are any other issues with using this scheme:
  
  $H_2(e(P, H(m))) \stackrel{?}{=} s$


The obvious problem is that anyone with the public key can compute everything on the left side, and hence forge a signature to any message they want.