How could one construct an algorithm for finding a node in a binary search tree that only requires the presence of $<$ on the key type. The ones I can easily also requires $=$.
If what you mean is that you want to build a BST and you only have the $<$ operation, and you only know the algorithms with the $\leq$ operation, you can notice that : $$a \leq b \Leftrightarrow \neg (a > b)$$ $$\Leftrightarrow \neg(b < a )$$ Hence, using a negation in the right place, you can build your usual algorithm.
You can test whether $a=b$ as follows: if it's not true that $a<b$ and not true that $b<a$, then it follows that $a=b$.
(Disclaimer: this requires that it be possible to order all of the elements using $<$, i.e., $\le$ be a total order. I imagine you were assuming this. But if it's not, you're screwed anyway and the problem is not solvable -- binary search trees require a total order.)