In my opinion, for the purpose of evaluating mesh convergence, it is possible to use the values for the Von Mises equivalent stress. The objective of this evaluation is to find out if the mesh is having some influence in your results. Since the von Mises´ equivalent stress is the combination of all principal stresses, if your mesh is not affecting the result of, let’s say, the max value of the von Mises stress in your model, probably, it is not affecting the stress distribution in general.
when you do this, keep in mind that if the shear is zero and the allowable of Tsai-Wu are the same of yield in von Mises then (von Mises/yield)^2 and Tsai-Wu failure index are the same. If not, pay attention since the behavior of composite is different and may differ according to the kind of fiber. For more information refere to WWFE (Worl-Wide Failure Excercise - http://www.sciencedirect.com/science/book/9780080444758).
Since the failure typically happens in inter fiber shear it may happen at much lower values than the Von-Mises stress. I would recommend identifying an approximate failure mechanism and based on that you can decide on the stress components that would likely drive the failure.
For the problem of convergence of finite element problems, I suggest you read:
R. B. Agarwal, Introductions to Finite Elements Analysis , ME 273 Lectures Notes http://www.engr.sjsu.edu/ragarwal/ME165/ME165_Lecture_Notes_files/FEA_Lectures/Chapter_1_%20Introduction.pdf , 2016
You will find alternative to the Von Mises stress (who will give a punctual convergence solution) using the total energy of deformation.
Von Mises strength criterion is used in good condition for metals. I do not know if it is contraindicated in composite materials, have searched the literature.
I hope the publication even indicated to help and give you an alternative.