Short answer
No. In classical mechanics there is no extra “collision-term’’ that must be added to the action of a gas.
If the molecules interact through a pair potential

  V(r) = 0   for r > 2R
  V(r) = ∞  for r ≤ 2R (hard spheres)

then along every admissible trajectory one always has r>2R, so V≡0 and the action of N particles is simply

  S[ri(t)] = ∫dt ∑i ½ mi |v i|² . (1)

Impulsive changes of the velocities at the instants of collision affect the time history of T = ½ mi v² but do not add any constant to the time integral.
If one replaces the hard core by a very steep but finite repulsive potential, V(r) is non-zero for a short time during each encounter; the integral ∫V dt for one collision is of order R Δp, depends on the relative velocity, and therefore on the particular microscopic path. It is not a universal constant that could be written as (2).

Why the derivation in the question fails

1. Action versus averages.
   The action principle is formulated for a single microscopic trajectory. Introducing “average force = pressure”, the “mean free path” λ, or the Boltzmann collision density mixes in equilibrium averages that are meaningful only after one has performed a statistical-mechanical ensemble average. They cannot be inserted into the microscopic Lagrangian.

2. A constant in the Lagrangian is arbitrary.
   One is always free to add an overall constant C to the Lagrangian, L → L − C. This shifts the action by C T (T is the time interval) but it has no effect on the Euler–Lagrange equations. Whether the gas is ideal, hard-sphere or Lennard-Jones, the freedom to choose the zero of potential energy remains; it is not generated by collisions and it certainly is not fixed by an expression such as (2).

3. Dependence on dynamical variables.
   The right-hand side of eq. (2) contains N, the derivative of the potential at 2R, the mean free path λ, the temperature (through ⟨μ⟩), etc.
   None of these are constants along an individual trajectory; they vary from realisation to realisation, so (2) cannot be a universal additive constant in the action.

Proper description of collisions in the action formalism

• Hard spheres: keep V = 0 outside the core and forbid trajectories with r ≤ 2R. The action is purely kinetic (eq. (1)); impulses are encoded in the discontinuities of v i(t).

• Soft potentials (e.g. Lennard-Jones): use the usual pair potential V(r); the action is

  S = ∫dt [ ∑i ½ mi v i² − ∑i<j V(|ri−rj|) ]. (2)

No additional term is required.

Consequences for statistical mechanics

In the (imaginary-time) path-integral representation of the partition function a constant shift of the potential multiplies Z by e^{-β C}, which simply shifts the zero of free energy. Hard-sphere corrections to the equation of state enter through the configurational integral of e^{-β V(r)}, giving the second and higher virial coefficients; again, no distinguished “collision contribution to the action’’ appears.

Conclusion

Collisions modify the trajectories and therefore the time dependence of the kinetic energy, but they do not produce an additive constant in the microscopic action. Equation (2) is not physically meaningful as a universal contribution to S.