I'll assume you are happy changing coordinates in $\mathbb{P}^2$ to whatever you want. I'll approach the complex geometry problem and ignore the number theory. Let you cubic be $f(x,y,z)$.

You write $f(x,y,1) = L_1(x,y,1) L_2(x,y,1) L_3(x,y,1) - 1$, for three linear forms $L_i$. Working homogeneously, you want $f(x,y,z) = L_1(x,y,z) L_2(x,y,z) L_3(x,y,z) - z^3$. In other words, you want to write $f$ as a linear combination of a product of three lines, and a triple line.

The space of all cubics is $\mathbb{P}^9$. (There are $10$ coefficients in a cubic, and we don't care about rescaling.)

The space of cubics which are triple lines is the $3$-uple Veronese embedding of $\mathbb{P}^2$ into $\mathbb{P}^9$. Calling the space of triple lines $V$; it has degree $3^2=9$ and dimension $2$.

The space of cubics which are a product of three lines is a finite ($6$ to $1$) projection of the Segre embedding of $\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^2$ into $\mathbb{P}^{26}$. Call the spaces of products of three lines $W$. The Segre embedding has degree $6!/(2! 2! 2!) = 90$, if I recall correctly. The Segre product stays away from the base points of the projection -- explicitly, you can't have three lines $(L_1, L_2, L_3)$ such that $L_1 L_2 L_3=0$. **Corrected from earlier version:** The map from the Segre product to $W$ is $6$ to $1$ (the $6$ orderings of the lines). So $W$ has degree $90/6=15$.

Morally, we have a point $x$ (our given cubic) in $\mathbb{P}^9$, and we want to know how many lines through $x$ meet $V$ and $W$. However, there is a problem. In fact, $W$ contains $V$! So there is a huge excess intersection contribution. The space of lines through $V$ and $W$ thus splits into two components: Lines which meet $V$ at one point and $W$ at another point; and just the space of all lines that meet $V$. We want to understand the first space. But separating out the second component is going to require an excess intersection computation which I'm not sure how to do. So I'll stop here.

If anyone wants to complete the computation, I'll leave this answer as Community Wiki.