Caloric equation of state for the Van der Waals gas

I’m currently trying to reproduce a specific derivation of the caloric equation of state for the Van der Waals gas, which I saw a couple of months ago in a Thermodynamics lecture. I’m well aware of the fact that there are already multiple derivations on this site and online (see [1], [2], [3], [4], [5]), but none of them are what I’m looking for.

The things that I remember from the derivation:

1. The inner energy was a function of $T$ and $v$.
2. We used the fact that $s$ is a state variable.
3. After using the first law we expanded all differentials in some variables.
4. With this we obtained $u(T)=f(T)-a/v$ and then proceeded to find $f(T)$ by using the fact that $C_V = (delta Q/partial T)_V = (partial u/partial T)_V$.

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What I’ve tried so far: Using the first law we obtain $Tds=du+pdv$. We can express $du$ and $dv$ in terms of $T,p$, which leads to the equation
$$ds = frac{1}{T}left(frac{partial u}{partial p}Bigg|_{T} + p frac{partial v}{partial p}Bigg|_{T}right)dp+frac{1}{T}left(frac{partial u}{partial T}Bigg|_{p}+ pfrac{partial v}{partial T}Bigg|_{p}right)dT.$$
Using the fact that $s$ is a state variable we get the relation
$$-frac{partial v}{partial p}Bigg|_{T} = frac{1}{T}left(frac{partial u}{partial T}Bigg|_{p}+p frac{partial v}{partial T}Bigg|_{p}right).$$
This simplifies the first equation a bit but isn’t really helpful since it’s impossible to acutally take the derivative of $v$ with respect to any variable due to the form of the Van der Waals equation.

Since I couldn’t figure anything out with this I tried the following as another approach:
$$begin{align} ds &= frac{1}{T} left(frac{partial u}{partial v}Bigg|_{T}dv + frac{partial u}{partial T}Bigg|_{v}dTright) + frac{p}{T}dv
& = C_V frac{dT}{T}+ frac{1}{T}left(frac{partial u}{partial v}Bigg|_{T}+p right)dv.end{align}$$
This approach seems to similar to the one from [1], but I’m 100% sure that we didn’t use the Helmholtz equation as suggested by @juanrga.

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TL;DR Could someone help me derive $u(T)=C_VT-a/v$ (caloric equaiton of state) for the Van der Waals gas without using any fancy equations, but rather fundamentals like the fact that $s$ is a state variable, etc.