No – the statement is already false in the smallest possible situation, namely for cyclic modules. A single counter-example is enough to show that the desired “algebraic saturation’’ property does not hold in general.

Notation. Throughout let 𝔬 be a DVR with maximal ideal 𝔭 and finite residue field 𝑘=𝔬/𝔭, and let the “type’’ of a finite 𝔬–module be the usual partition recording the exponents of its cyclic direct summands.

1. A counter–example with n = 2

Take

 λ = (2), μ = (1), ν = (1)    (partitions of length 1).

For partitions of length 1 the Littlewood–Richardson coefficient is just the ordinary sum, so

 c^{λ}{μ,ν}=c^{(2)}{(1),(1)} = 1. (∗)

Consequently, for every positive integer n we still have

 c^{2λ}{2μ,2ν}=c^{(4)}{(2),(2)} = 1,

so the hypotheses in the question are satisfied when n = 2.

Now work over any DVR 𝔬 (e.g. 𝔬 = ℤ_p).
Put

 M = 𝔬/𝔭⁴. (type 2λ = (4))

Choose the submodule

 N = 𝔭²/𝔭⁴ ⊂ M , (type 2μ = (2))

so that M/N ≅ 𝔬/𝔭² has type 2ν = (2).

We must ask whether there is a submodule

 L ⊂ M of type λ = (2)

such that

 L ∩ N has type μ = (1) and (L/(L∩N)) has type ν = (1).

But M is cyclic of length 4, hence for each 0 ≤ t ≤ 4 it possesses exactly one submodule of length t, namely 𝔭^{4-t}M. The unique submodule whose type is (2) is therefore

 L = 𝔭²M = 𝔭²/𝔭⁴.

Hence L coincides with N, so

 L ∩ N = N  (type (2)),

whereas we would need this intersection to have type (1). Thus no such L exists, even though the Littlewood–Richardson coefficient required by the Hall–criterion is non–zero.

2. Why no choice of L can work in a cyclic module

More generally, let a ≥ 2 and consider

 λ = (a), n = 2, μ = (b), ν = (a−b)  (1 ≤ b ≤ a−1).

Again c^{λ}{μ,ν}=1 and therefore c^{2λ}{2μ,2ν}=1.
Take M = 𝔬/𝔭^{2a} and N = 𝔭^{2a−2b}M (type 2μ = (2b)). The only submodule of M whose type is λ is L = 𝔭^{a}M. An easy valuation calculation shows

 L ∩ N = { 𝔭^{max(a,2a−2b)}M } ,

so L ∩ N has type (2b) if 2b ≤ a and type (a) if 2b ≥ a; it is never of type (b) unless b=0 or b=a. Hence the desired refinement fails for every intermediate b.

3. Conclusion

Even in the one–row (i.e. completely cyclic) case the strengthened statement is false, so in general one cannot expect to extract a submodule of type λ whose intersection with a given nμ–submodule has type μ.

In other words, the Hall–module realisation of Littlewood–Richardson coefficients does satisfy saturation of the numerical coefficient

 c^{nμ,nν}{nλ} ≠ 0 ⇒ c^{μ,ν}{λ} ≠ 0,

but it does not admit a corresponding “geometric’’ or “module–theoretic’’ lifting in the sense asked for in the question.