Graham Priest promotes a wildly incorrect view of Buddhist logic and teachings. He believes that Buddhist doctrine contains contradictions, which is utter nonsense. His problem is that he does not properly understand Aristotle's logic so cannot understand Nagarjuna. Not being a Buddhist, Priest has no grasp of the Two Truths doctrine and seems to believe it leads to contradictions, which is such a complete misunderstanding it seems almost wilful.
Let's examine the question here.
..."Western/Aristotelian logic doesn't tolerate contradictions (principle of non-contradiction)..."
..."and everything has to be either true or false (principle of the excluded middle),
This is wrong. The principle of bivalence is what you're thinking of and this is not necessary for dialectical logic. This logic works with statements that are true or false (in the form A/not-A) but it does not say that all statements must be true or false. The LEM is applicable where a pair of statements meet Aristotle's specification, (which is that one is true and one false), but statements do not have meet this specification. Where they do not the law does not apply. This is the point that Priest misunderstands, and it is why Buddhist doctrine contains no contradictions.
..." but Buddhist logic follows a system called the catuskoti which implies that statements can be true, false, true and false, or neither true nor false."
This is an utterly ridiculous idea. I can't imagine how anyone could arrive at it. Buddhist logic is no different from Aristotle's, it's just that it is applied to pairs of statements on two axes (this or that, neither or both). Priest muddles the issues and creates complications where none are necessary. He does Buddhism a massive disservice, as is shown by the low view you have formed of Buddhist logic after reading him. He seems to believe that Buddhists are idiots.
I would suggest that Priest is ignored. Nagarjuna offers us a rigorous application of Aristotelian logic and if it were not rigorous the whole argument in Fundamental Wisdom would fall apart.
For a longer response I have an essay on Bernardo Kastrup's blog entitled, 'Aristotle, Nagarjuna and the Law of Non-Contradiction in Buddhist Philosophy'. https://www.bernardokastrup.com/2017/05/aristotle-nagarjuna-and-law-of-non.html
This puts paid to Priest's anti-Buddhist ideas. He should take the trouble to study Buddhism.
There are NO dialectical or 'true' contradictions in Buddhist doctrine and this is exactly what Nagarjuna proves.