How to simplify a the signing process in an elliptic curve signature scheme that involves a quadratic verification equation?

I am looking at a proprietary signature scheme used in production. It involves a short Weierstrass curve $E_{mathcal{W}}:y^2=x^3+ax+b$ in the prime field $mathbb{F}_p$. The parameters are set up such that $E_{mathcal{W}}$ is always expressible as a Montgomery curve $E_{mathcal{M}}:y^2=x^3+x$ (i.e. $a_{mathcal{W}}=1$, $b_{mathcal{W}}=0$, $A_{mathcal{M}}=0$, and $B_{mathcal{M}}=1$). As far as I know, the Montgomery form is never used for verification. The curve has highly composite order $n$, with a base point $B$ having prime order $ell$.

The verification process given a hash function $H$, a keyed hash function built from $H$ with (namely $H$ in HMAC mode, $H_k$), a message $M$, a public key $K$ and a signature consisting of a scalar $s$ and a hash $h$ is performed as follows:

1. $h_1=H_{c_1}(M||h)$.
2. $R=scdot(sB+h_1K)$
3. $h_2=H_{c_2}(M || R_x || R_y)$, where $R_x$ is the $x$ coordinate of $R$ and accordingly $R_y$ is the $y$ coordinate of $R$
4. If $h_2=h$, the signature is valid; else, it is invalid.

$c_1$ and $c_2$ are static HMAC keys known both to the signer and the verifier. My conjecture is that they act as domain separation strings.

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I am trying to determine if there is an efficient way of creating a signature that does not involve taking a square root in $mathbb{F}_p$. Square roots are not trivially found in $p$ because $p$ it may be that $pequiv1pmod{4}$ and $pequiv1pmod{8}$.
Currently, I reach the following signing process:

1. Choose a nonce $r$ such that $0<r<ell$.
2. $R=rB$
3. $h_2=H_{c_2}(M||R_x||R_y)$
4. $h_1=H_{c_1}(M||h_2)$
5. $s=frac{-Hkpmsqrt{(h_1k)^2+4r}}{2}pmod{ell}$, where $k$ is the secret key corresponding to $K$ in the verification process
6. If $sqrt{(h_1k)^2+4r}$ has no solution in $mathbb{F}_p$, restart from the beginning.
7. Output signature $(s, h_2)$.

Is there a way to create a signature passing the above verification process that does not involve a square root in $mathbb{F}_p$?