It's been shown to be surpassed at n=150, which as you note is likely very generous. Hypertree would typically only require a few more states. Hypertree doesn't grow meaningfully faster than Tree. Hypertree(3) is what would be called a Salad number, combining a very fast growing function with some very weak one(s) such as iteration, which in the Fast Growing Hierarchy corresponds to taking the successor if an ordinal.
BB(n) surpasses Tree(n) by - at most - when n=2645.
And likely shortly after BB(100).
Now consider the following definition for an exponentially faster growing number:
HyperTree(n) is where the Tree function is nested x times, where x is the result of tree(n).
HyperTree(3) = Tree(Tree(Tree..{Tree(3) long}...Tree(3))))...)
BB(X) would (should) still outpace HyperTree(x) at some value of x.
I don't know where to begin even to give an upper or lower bound of when BB(x) is larger than HyperTree(x).