Invariant perturbation theory in quantum field theories were not first derived using interaction picture and Dyson series, Feynman derived them from a path integral, Schwinger from an action principle, while Stueckelberg used his own methods which were derived from (so-called) Schwinger-Dyson equations and canonical commutation relations, which are the content of the path-integral rewritten as differential equations. All these methods are manifestly invariant, and do not talk about the unitary relation between free and interacting vacuum.

Perturbation theory can be derived using interaction picture, but this derivation is hokey, because the unitary relation between free and interacting fields is not well defined in the infinite volume limit. You can perform the calculations with a finite volume, and take the limit explicitly at the end to see what is going wrong.

The path integral defines the perturbation series without reference to any free-particle vacuum. The path-integral is well defined in Euclidean space with only an ultraviolet regulator, and it does not have a Haag problem, in that it doesn't need to worry about relating the interacting vacuum to a free vacuum by a unitary map. The relation between path-integral correlation functions and scattering states is given by the LSZ construction, and there are no issues with vacuum bubbles, they factor out.

Haag's theorem is really nothing deep--- it's the statement that in any realistic quantum field theory, the true vacuum always has an infinite number of particles when compared to the bare vacuum. In a box, the true vacuum becomes orthogonal in the limit to the bare vacuum, because the amplitudes to leave the vacuum grow vanishes as the square root of the volume, and the new vacuum has an expected density of particles, so that the probability that there are zero particles exactly (the inner product with the Fock vacuum) vanishes exponentially fast . The reason there are always matrix elements out of the Fock vacuum is that when you can scatter, then you can pair-produce (by crossing), and this means that the inner product of the interacting vacuum with the bare vacuum is strictly less than one, because it also includes two particle states. But the production rate of particles in the bare-vacuum is per-unit-volume (by translation invariance), and if it is nonzero in any volume, it is infinite, so the inner product must vanish. You can't fix this, it doesn't depend on ultraviolet details, it's just a property of large-volume. The true vacuum always contains an infinite number of particles defined relative to the bare vacuum. Surprisingly, this part of the argument is reviewed ok in the linked Earman article, except verbosely.

The reason the interaction picture doesn't care about the Haag issue is just because it is not describing the transition probability between bare and interacting vacuum, it is describing the transition probability for scattering. In the Dyson-style interaction picture, there's a turning on-and-off function f(t) that describes the interaction strength. This turning on and off is supposed to be adiabatic, so that the bare vacuum slowly relaxes into the interacting vacuum, the scattering happens, and then you turn off the interaction again.

If you were to ask, in the interaction picture, what is the probability for the vacuum to remain the vacuum between any two time slices with different values of f(t), this amplitude includes a volume integral, this second order term will produce the same infinite transition rate, and show you that the physical vacuum is completely orthogonal at any two times.

Who cares, really. You never ask this question. You are interested in scattering of physical particles. Here, you are looking at excitations, and you jiggle the interaction Hamiltonian to fix their interactions (and the vacuum energy) by renormalization prescription. You could imagine doing all this in a box, where the adiabatic prescription for the Dyson turning on and off makes sense.

To see that Haag's theorem is not an artifact, or nonsense, that it's physical, consider the situation in QCD. Here, the noninteracting vacuum consists of free quarks and gluon excitations with a small scattering. But at any infinitesimal coupling, you will eventually churn the vacuum completely, so that a free quark state gets an infinite mass! In this case, the infrared behavior clearly takes the free quark states and free gluon states right out of the Hilbert space of the theory.

This doesn't happen in a finite volume where you adiabatically turn on the interaction. The ultimate explanation for why Dyson's derivation works well is that it is justified in an appropriate regulator: it is correct in finite volume box. Dyson is considering quantities such as scattering quantities which have a sensible limit as the box is made big. He isn't considering the H-atom, which only sticks around as long as the coupling is turned on.