Why doesn't Dark Matter participate in BBN reactions?

This is a slightly different variation on an earlier question:

In the very early universe, the hot plasma consisted of fixed amount of radiation (photons and neutrinos) and matter (electrons, protons, neutrons, etc). There were many competing reaction taking place and using statistical methods I understand that you can deduce the particle content of the universe when radiation and particles began to condense into nucleons (at roughly 100 Giga K).

Dark Matter interacts weakly with matter (this, as I understand it, is the basis behind the LUX experiment) as do neutrons.  Neither Dark Matter nor neutrons interact with matter using electromagnetic forces.  Neutrons appear to factor into several of the reactions in Big Bang Nucleosynthesis (BBN) yet Dark Matter is at least five times more abundant than neutrons.  So why doesn't Dark Matter participate in the BBN reactions with roughly the same probability as neutrons?

• Neutrons do not interact weakly (compared with dark matter). – Rob Jeffries Oct 18 '16 at 22:15
• How do protons and neutrons react to form deuterium, then? – Quarkly Oct 18 '16 at 22:57

Neutrons do feel both electromagnetic (they have a magnetic moment) and of course the strong nuclear forces. Their cross-section with matter is much higher than any conceivable interaction for WIMPS that may only interact through the weak force. At the time of primordial nucleosynthesis the matter density of the universe was less than 1 kg/m$^3$, so there is no reason to suppose that dark matter would interact significantly.
In the Sun, the fusion that produces a deuteron is via a weak force interaction after proton fusion. In big bang nucleosynthesis neutrons and protons interact directly ("neutron burning") via a radiative capture reaction involving the strong and electromagnetic forces. $$p + n \rightarrow D + \gamma$$ The velocity-averaged cross-section for this reaction at the typical 80 keV temperature of primordial nucleosynthesis is a few$\times 10^{-32}$ m$^{2}$ (e.g. Gorbunov & Rubakov 2011, p.152)$^*$.
The upper limits on the cross-section for nucleon interactions with a few hundred GeV WIMPS (this would still be "cold" dark matter, even at the epoch of nucleosynthesis) is $<10^{-49}$ m$^2$. (http://luxdarkmatter.org/).
$^*$ Fortunately there are not (many) free neutrons in the Sun!