The "inner" observer has observed the outcome of the experiment well before the "outer" observer but there is no contradiction whatsoever.
In particular, if the "outer" observer properly defines the projection operator that determines whether the "inner" observer had a well-defined feeling or perception of knowledge (with some allowed error margin for confusion) after he did his measurement, the "outer" observer will find out that indeed, the "inner" observer had a well-defined feeling. So they will surely agree about that. However, the "outer" observer couldn't have predicted which of the outcomes was actually measured by the "inner observer": he can just prove that the "inner" observer had a well-defined perception.
However, the outer observer is allowed - and encouraged - to use quantum mechanics uniformly for all objects, including bound states of protons and electrons that are sometimes called "other people". If he does so, the wave function - encoding the probability amplitudes sufficient to make any future measurements - is uniformly spreading and linear superpositions of macroscopically distinct states are omnipresent as the intermediate states.
This was discussed nicely e.g. in Sidney Coleman's "Quantum Mechanics In Your Face"
Let me mention that quite typically, the "inner" observer could be "less reliable" and "less decoherent" than the outer one. So he could "perceive" that some other decohered histories have already disappeared as possibilities, but the interference could get restore later etc. which could ultimately refute the "inner" observer's classical memory. That's why a more reliable, more decohered "outer" observer shouldn't be imagining that the wave function of the observer system collapsed before the measurements are actually done - and/or perceived - by the "outer" observer.
This comment becomes completely obvious if you imagine that the inner observer is just a collection of a small number of atoms that is decohering pretty slowly. Still, some perception of classical outcomes and decoherence may start to emerge. But it's still fuzzy enough for the classical approximation to be inaccurate. It's clear that in such a situation, the outer observer - who is much more reliably "classical" - can't believe the would-be classical concepts used by the "inner" observer because the "outer" observer knows very well that the "inner" observer is just a quantum mechanical system for which the classical approximation is highly imperfect.
I think that the reason why you and others have problems with these simple things is that you still fundamentally refuse to accept that at the fundamental level, the world is quantum mechanical, and that all classical concepts - such as well-defined positions or completely well-defined results of experiments that become "absolute facts" - are just approximate. They are approximate because classical physics is not the way how the world works at the deepest level. If one wants to describe any system totally accurately, he has to use the full quantum mechanical theory where everything is allowed to have a nonzero chance to interfere and "recohere", whether or not other "conscious beings" are parts of this world.
At the same moment, every observer who uses a quantum mechanical description and who wants to convert some observations into "facts" must make sure that the "facts" are mutually decohered or "consistent" histories. Still, everyone should acknowledge that all such facts are always an approximation assuming that decoherence is sufficiently strong to make it de facto impossible for the historical facts to be "reverted" or "retroactively modified" by the future evolution. Quantum, small enough particles do many non-classical things. Quantum tunneling is an example. A small observer's perception that a measurement has become a "fact", but this fact is later "undone" by subsequent recoherence, is another example. That's why one must never assume "too early" that some components of the wave functions have disappeared.
By the way, you also incorrectly suggest that the lab with the inner observer may be "screened against decoherence". This is just a misunderstanding of what decoherence means. Decoherence depends on the separation of the degrees of freedom of the studied system and the environmental degrees of freedom. But that doesn't mean that the "environment" has to be physically separated from the "interesting system". Quite generally, many of the environmental degrees of freedom are geometrically located "inside" the objects we are interested in. The environment is composed of all the useless, "high-entropy" information we're not interested in - surely things like lots of infrared photons traveling through our bodies, reflecting from the walls of the lab, and so on. You may isolate the lab from the external world but you will not stop the fact that it's still valid to describe the phenomena inside the lab by decoherence.
If the "outer" observer knows how to define the degrees of freedom in the lab to the interesting and environmental ones, he will also find out that the different states of the "inner" observer's brain have quickly decohered from each other. That means that for the purpose of all predictions about the "inner" observer only, the quantum probability waves may be replaced - with a good enough accuracy - by a nearly diagonal density matrix. The diagonal entries are the probabilities but again, the "outer" observer is not forced to imagine that the (later unobserved) a priori possible outcomes have seen their density matrix elements collapse to zero. He naturally does it only when he actually makes the measurement at the very end. When the decoherence happens in the middle, and it's reliable enough, it's consistent for the outer observer to assume that one of the results was "already real" after the "inner" observer made his measurement. But the "outer" observer surely doesn't have to assume that such a premature collapse has taken place. Why would he do it? Because of the respect to the "inner" observer as a fellow human with consciousness? No such "respect" exists in science. Science doesn't care whether someone may claim to be an observer - it still allows accurate predictions for any conglomerates of particles.