Schnorr signature verification via Pairing-Based Cryptography?

Let $$e: G^{\dagger}_{1} \times G_{2} \mapsto G_{T}$$ define the pairing. Assuming an output $$\sigma = (c, p)$$ similar to a Schnorr signature:

1. $$s \times Y \mapsto Y_{s}$$ and $$m \times G \mapsto M$$
2. $$c = H(Y_{s}||M||B)$$
3. $$p \times Y = M + c \times Y_{s}$$

However, with additional requirements:

1. $$p$$ is a static secret, where $$p \times G^{\dagger} \mapsto P^{\dagger}$$ and $$P^{\dagger}$$ is public.
2. $$Y$$ is a base point that can change such that $$y = m \cdot (p - c \cdot s)^{-1}$$ and $$y \times G \mapsto Y$$.

Can we use PBC in Symmetric-XDH settings for an alternative verification only giving the result $$Y$$ and knowing $$(Y_{s}, M, B)$$ ?

1. $$c = H(Y_{s}||M||B)$$
2. $$e(P^{\dagger}, Y) \stackrel{?}{=} e(G^{\dagger}, M) \cdot e(G^{\dagger}, Y_{s})^{c}$$

If we assume that $$P^{\dagger}$$ is static forcing also $$p$$ to be static. Is there a way to forge such scheme?

For an adversary, there is a sequence of events where he must know $$(Y_{s}, M)$$ to derive $$c$$. But those must be selected to result in $$p \times Y$$ !