Term for binary search tree using hashes?
To answer the explicitly asked question: There probably is no specific name for this data structure since noone has bothered to use or study it. 

Regarding the underlying question: Even using a "good" hash function, you can not expect as well a degree of balanced as you think: 

Looking at the node with key $foo$, we see that all keys (in the subtree below $foo$) with a hash smaller than $h(foo)$ will be in the left subtree, while all those with a larger hash will be in the right subtree. So the balancedness of the tree will depend on how far $h(foo)$ is away from the center. This has an expected value of $\frac 12 N$, where $N$ is the number of hash values that belong in the subtree below $foo$. So we can expect an average weight balance of $1:3$ in each node. While this is still enough to give logarithmic expected height, it is already notably worse than the $1:1.6$ balance an AVL tree has in the worst case. Factoring in the additional risk of the hash function being not "good" for the data of a specific run, the idea quickly becomes unattractive.

Additionally, in many practical scenarios you have the option of using a library implementation of AVL trees, while you have to implement the "hash tree" yourself. So your "for free" argument will typically be the other way round.