Haag's theorem is a result about ground states (more generally, pure states that are invariant under the Poincaré group) and their induced GNS representation of canonical (anti)commutation relations.
It says that two invariant states are either equal or disjoint, where disjoint means that there is no unitary map relating the two corresponding irreducible representations of the algebra of canonical spacetime fields.
The physical consequence is that free and interacting theories, having usually distinct ground states (vacua), are represented in inequivalent ways (do not have "the same Hilbert space", stated in a rougher way).
In order to do scattering theory, in particular to write the S-matrix, one has therefore to do something different from what the naïve ideas suggest. In principle, this could be a serious obstacle to a mathematically rigorous approach to perturbative QFT, but actually it has already been overcome long ago by Haag himself. The so-called Haag-Ruelle scattering theory (for a reference, one may consult the books by Reed-Simon, vol 3 or 4 I don't remember, and references therein contained) takes into account this problem successfully: an S-matrix is written, and also LSZ formulas are proved.
Of course in Haag-Ruelle scattering theory it is assumed that both the free and the interacting theories are well-defined in a rigorous way (i.e. satisfying the so-called Wightman axioms, or the Osterwalder-Schrader euclidean version).
I am not familiar with the so-called causal perturbation theory. But to me it seems just a way to try and understand the correlation functions of the interacting theory, knowing which the whole theory can be reconstructed (as shown also by Wightman). This is the aim of constructive QFT that was developed mostly in the 60s-70s by Glimm, Jaffe, Simon and others. Using this approach, some interacting theories could be well-defined, such as $P(\phi)_2, \phi^4_3$ and Yukawa in 3 spacetime dimensions. However, the relevant theories in 4 spacetime dimensions have not yet been rigorously dealt with.
To sum up, once the interacting theory is known rigorously, either by tools of causal perturbation theory or by other means, then applying Haag-Ruelle scattering theory would overcome the mathematical difficulties of having inequivalent representations and allows to construct the S-matrix and derive the LSZ formulas, thus putting on solid grounds the perturbation theory used by physicists. Of course, one may see things turned around, and say that the success of predictions based on perturbation theory assures that there is a rigorous, unambiguous, underlying theory. I am sure that this was and still is the point of view driving people to study the rigorous aspects of QFT.