There is no better upper bound. Suppose that your torus is the factor of the plane by integer lattice, and the metric is Euclidean. Let $a=m$ and $b=m+1$. These generate the lattice. And length of $ab$ is $\sqrt{4m^2+1}\sim 2m, m\to\infty$.

There is no better upper bound independent of the area. Suppose that your torus is the factor of the plane by integer lattice, and the metric is Euclidean. Let $a=m$ and $b=m+1$. These generate the lattice. And length of $ab$ is $\sqrt{4m^2+1}\sim 2m, m\to\infty$.

If you want to include area, you can obtain an exact formula, rather than an inequality. Denote the angle between $a$ and $b$ by $\phi$, then $L^2(ab)=L^2(a)+L^2(b)+2L(a)L(b)\cos\phi,$ while the area $A=2L(a)L(b)\sin\phi$. Eliminating $\phi$ we obtain $$L^2(ab)=L^2(a)+L^2(b)+2\sqrt{L^2(a)L^2(b)-A^2}.$$