If $\dfrac{4x^2-1}{4x^2-y^2}$ is an integer, then it is $1$

This is a fun problem! Where did you find it?

There are no solutions except for $k=1$. Assume from now on that $k \neq 1$. Since $k$ is clearly odd, it is also not $0$ and we deduce that $k(k-1)>0$. It is convenient to set $M = k(k-1)$. Also, we may assume WLOG that $x$ and $y\geq0$.

As you did, rewrite the equation to $$ky^2 - 4 (k-1) x^2 = 1$$ or $$(ky)^2 - k(k-1) (2x)^2 = k.$$ Set $Y=ky$ and $X=2x$ so the equation is $$Y^2 - M X^2 = k. \quad (\ast)$$

We will study the equation $(\ast)$. In the end, we will see that there are no solutions with $X$ even and $Y$ divisible by $k$.

The rest of this proof works inside the ring $R:=\mathbb{Z}[\sqrt{M}]$. Note that $M=k(k-1)$ is not square, so this is an integral domain. For an element $\alpha = a+b \sqrt{M} \in R$, set $\bar{\alpha} = a - b \sqrt{M}$.

Set $\epsilon = (2k-1) + 2 \sqrt{M}$. Note that $\epsilon \bar{\epsilon} = (2k-1)^2 - 4 k(k-1) = 1$, so $\epsilon$ is a unit of $R$.

Set $\delta = Y+X \sqrt{M}$. Since $\delta$ is a positive real, and $\epsilon>1$, there is some integer $n$ such that $\epsilon^n \leq \delta < \epsilon^{n+1}$. Write $\delta = \gamma \epsilon^n$. Since $\epsilon$ is a unit, $\gamma$ is in the ring $R$ and, by construction, $$1 \leq \gamma < \epsilon.$$ We have $$\epsilon = 2k-1 + 2 \sqrt{k(k-1)} < 2k+ 2k = 4k.$$ So $$1 \leq \gamma < 4k.$$

But, also $\gamma \bar{\gamma} = \delta \bar{\delta} =k$. So $$\frac{1}{4} \leq \bar{\gamma} \leq k.$$ Write $\gamma = U + V \sqrt{M}$. So $$\begin{matrix} 1 & \leq & U+V\sqrt{M} & < & 4k \\ \frac{1}{4} & \leq & U-V\sqrt{M} & < & k \\ \end{matrix}.$$ Solving for $V$, we have $$\frac{-k}{2 \sqrt{M}} < V < \frac{2k}{\sqrt{M}}.$$ The LHS is $\approx -1/2$, and the right hand side is slightly larger than $2$. So $V$ is $0$, $1$ or $2$. We break into cases:

$\bullet$ If $V=0$, then the equation $\gamma \bar{\gamma} =k$ gives $U^2 =k$. So $k$ is a square, say $k=m^2$ and $M=m^2(m^2-1)$. We have $$Y+X \sqrt{M} = m ((2k-1)+2 \sqrt{M})^n = m (2m^2-1 + 2 m \sqrt{m^2-1})^n.$$ An easy induction on $n$ shows that $Y \equiv (-1)^n m \mod m^2$ so, except when $m=1$, we do not have $Y$ divisible by $k=m^2$. Of course, the case $m=1$ corresponds to $k=1$.

$\bullet$ If $V=1$ then $U^2 - k(k-1) = k$ so $U=k$. We have $$Y+ X \sqrt{M} = (k+\sqrt{M}) \cdot ((2k-1)+2 \sqrt{M})^n.$$ An easy induction on $n$ shows that $X$ is odd.

$\bullet$ If $V=2$ then $U^2 - 4k(k-1) = k$ so $4k^2-3k = U^2$. This gives $64k^2 - 48 k + 9= 64 U^2+9$ or $(8k-3)^2 - (8U)^2 = 9$. The only ways to write $9$ as a difference of squares are $3^2-0^2$ and $5^2-4^2$. The former gives $k=0$, but we saw that $k$ must be odd; the latter gives no solution since $8$ does not divide $4$.