If aggregate demand falls dramatically, so do prices. And if prices drop, so does demand for your “loanable funds”, and the interest rate need not rise above the “natural rate”.
Now, obviously nothing happens instantly. But pointing this out and labeling it a problem is a non-sequitur. If things were to happen instantly, then there would be no entrepreneurship and no market process. The changes you describe in your example, such as lower aggregate demand, is what will induce entrepreneurs to bid the interest rate down to compensate for a lower nominal reservoir of savings that also resulted from the increase in cash holdings, as per your example. And it is their entrepreneurial action that will adjust the interest rate and eventually all other prices. Not the other way around, i.e, prices adjust and then action is taken. This is what your theory implies - that entrepreneurs are unable to calculate until monetary equilibrium is reached. This is totally incompatible with Austrian price theory.
The important point to understand here is that the changes in economic activity, such as voluntary increase in demand for money, is by definition, aligned with changes in consumer preferences, so that signals are guiding entrepreneurs in the right direction. It is nonsense to say that there is a problem of “above” or “below” some natural rate, like in the case of monetary inflation (or deflation). The latter are temporary changes in demand brought about by force and not by consumer change in preferences. Like some kind of a concealed subsidy, they do not reflect any permanent changes in consumer preferences. It is only the latter [temporary coercive activity] that guides entrepreneurs astray.