I don't expect this to be true in general: let me clarify this point in the OP. (For the record, I've in mind some topological monoids considered by J. Snellman in the context of factorization theory.)

True. So let us restrict to "sufficiently small" topological monoids. What happens?

You should definitely give a look at A. Geroldinger and F. Halter-Koch's monograph on the factorization theory of (abelian cancellative) monoids: Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics 278, Chapman & Hall/CRC, 2006.

@Pietro Majer. I agree that the standing assumptions and $\dim(V)\ne 0$ imply $V\cap{\rm bd}(P)\ne\emptyset$, in view of the following: "If $C$ and $S$ are, resp., a compact and a connected unbounded subset of $\mathbb R^n$ s.t. $C\cap S\ne\emptyset$, then $S$ meets ${\rm bd}(P)$" (which in turn is an easy consequence of Proposition 3, Ch. I, Sect. I.1 in Bourbaki's General Topology: Part 1). I also agree that the bd of a polytope is the union of its proper faces. Yet, I think this is not enough for your induction to work. That said, I'm really looking for a reference, not for a proof.

For the record: An alternative proof that any free group (and hence any free monoid) is linearly orderable can be found in K. Iwasawa, On linearly ordered groups, J. Math. Soc. Japan 1 (1948), 1-9. Yet, Iwasawa's approach doesn't help much with the OP (as far as I can tell).

As for free groups on finite alphabets, see Section 5 in: D. M. Kim and D. Rolfsen, An Ordering for Groups of Pure Braids and Fibre-type Hyperplane Arrangements, Canad. J. Math. 55 (2002), 822-838 (and the references therein). I don't know if the proof of the general statement: "All free groups are linearly orderable" by the same method (which is as simple or difficult, it is up to you, as the finite case), is explicitly written down somewhere. Does anybody know?

Are the ratios $\sigma_{X}(k)/k$ eventually non-increasing for $X = {\rm Sp}(n)$ or whatever?