What observable are particle detectors really measuring during scattering and what really are particles mathematically?

If QFT naively worked like QM, one would expect the answer to my question to derive from this:

1. At the beginning of scattering, we prepare a state that is a smearing of eigenstates of the spatial integral of the free Hamiltonian density. E.g. this state can look like $|\psi_1 \rangle=\int dp_1 p_2 \psi (p_1) \phi(p_2)|p_1,p_2\rangle$
2. Time evolution of the state is generated by the spatial integral of the full Hamiltonian density.
3. The measurement device measures the operator corresponding to the spatial integral of the free Hamiltonian density. The post-measurement state can once again be something like, let's say, $|\psi _2\rangle = \int dp_1p_2dp_3 \psi(p_1,p_2,p_3) |p_1,p_2,p_3\rangle$

I have taken an example above in which we collide two particles and get three particles as product.

The answers to my titular question in this naive framework would have been :

What observable are we measuring?- Answer - $\int H_0(x)d^3x$

What particles are mathematically?- Answer - They are smearings eigenstates of the free Hamiltonian having a definite particle number. E.g. the phenomenology of something like $|\psi _1\rangle$ above corresponds to two particles in the real world lab whose incoming momenta are encoded in $\psi (p_1)$ and $\phi (p_2)$ respectively.

Problems with the naive framework:

1. A real world particle detector cannot correspond to an observable like $\int H_0(x)d^3x$ as this would violate causality as the integral is over the entirety of space. The observable that a real world particle detector corresponds to must be a local observable.
2. It doesn't make sense to talk about a state that is a smearing of eigenstates of the free Hamiltonian and then to evolve that state using the full Hamiltonian. The free Hamiltonian doesn't exist on the Hilbert space on which the full Hamiltonian lives. This is a consequence of Haag's theorem.

Suppose, for the sake of this question, that we have a Hilbert space representation of the CCR of some interacting QFT. Also assume that we have a full account of the Hilbert space in terms of the eigenvectors and eigenvalues of some CSCO which includes the full Hamiltonian operator.

If we had this full comprehensive account of the interacting QFT, how would we answer my titular question? I will re-iterate the problem in more detail :

1. What kind of operator in this comprehensive account of the interacting QFT actually corresponds to real world particle detector devices that we have in the LHC? The answer to this must be some local observable that can be constructed using the operators in the interacting theory.
2. What are particles, mathematically speaking, in this full comprehensive account of the interacting QFT? If you prepare some particles at time $t_i$ to collide them, what kind of state represents that initial state in this Hilbert space?

P.S.1 A guess for the answer to 1 (Just ignore this if it's wrong) : If our $n$ particle detector devices are located in regions $R_i$ of the lab, $i=1,2..n$, then the corresponding observables are $O_i= \int_ {R_i}H(x) d^3x$ where $H$ is the full Hamiltonian. The momentum observables corresponding to the detectors are $P^j_i=\int _{R_i} P^j(x)d^3x$. The reason for this guess is what happens classically : If you have a bunch of detectors in a room that is full of rays of light bouncing off the walls, an individual detector will only able to read the energy or momentum of the light rays that happen to hit it, i.e. the device is only able to read the energy-momentum in the region it resides.

P.S.2- If one takes things like Unruh effect to relativise the definition of a particle, the answer to (2) would depend on the frame of reference.

P.S.3- I am not making idealisations like "lab size tends to infinity" or $t_i\to -\infty$ because I want to understand what a completely detailed account of scattering would hypothetically look like. I want to understand what mathematical objects are our particles and what are our observables in the full bare bones account of scattering.