This is to elaborate on the comment I made above about Livingston's paper.

Livingston also proves (Theorem 4) that if a knot is null-homologous in a fiber surface of a torus knot and it bounds a quasipositive surface, then $\tau(K)=g(K)=g_4(K)$, where $g(K)$ is the three-genus and $g_4(K)$ is the four-genus. As pointed out by Hedden, this is equivalent to saying that if $K$ is *strongly quasipositive*, then $\tau=g(K)=g_4(K)$. Hedden also proves that for fibered knots the converse is true, i.e. $\tau(K)=g(K)=g_4(K)$ implies $K$ is strongly quasipositive.

So if you have some kind of Floer-homology-free way of figuring out whether your knot is fibered and (if it is) what its fiber surface is like (i.e. what's the three-genus? and is it strongly quasipositive?), that might be used to tell you $\tau(K)$.

As for figuring out the genera and fiberedness of your knot: in Gabai's paper ``Detecting fibered links in $S^3$" he gives a general strategy (using his sutured manifold theory) for figuring out whether any link in $S^3$ is fibered and what the fiber surface is like. Gabai classified the fiberedness and genera of pretzel knots and Hirasawa and Murasugi used Gabai's approach to do the same for Montesinos knots. I think you might be able to find similar info about arborescent knots. Apparently Stoimenow also has a paper classifying the strong-quasipositivity and fiberedness of closed 3-braids as well.

In that paper of Hedden's mentioned above he also connects the dots to contact topology --- in particular if your knot is fibered and the corresponding open book decomposition supports a tight contact structure, Hedden tells you that $\tau(K)=g(K)=g_4(K)$ as well. In all these approaches you need to be able to determine genus, which is not easy.