Is it possible to start fire using moonlight?

Moonlight has a spectral peak around $650\ \mathrm{nm}$ (the sun peaks at around $550\ \mathrm{nm}$). Ordinary solar cells will work just fine to convert it into electricity. The power of moonlight is about $500\,000$ times less than that of sunlight, which for a solar constant of $1000\ \mathrm{W/m^2}$ leaves us with about $2\ \mathrm{mW/m^2}$. After accounting for optical losses and a typical solar cell efficiency of roughly $20\ \%$, we can probably hope to extract approx. $0.1\ \mathrm{mW}$ with a fairly simple foil mirror of $1\ \mathrm{m^2}$ surface area. Accumulated over the course of a whole night with a full moon, this leaves us with around $6\ \mathrm h\times3600\ \mathrm{s/h}\times0.1\ \mathrm{mW}\approx2\ \mathrm J$ of energy. That's plenty of energy to ignite a fire using the right chemicals and a thin filament as a heater.

At least one point in your favour is that the light we receive from the Moon has barely anything to do with its temperature. Instead it is mostly a secondary light source "reflecting" light from the Sun towards us.

The second point in your favour (I think) is that the thermodynamic argument seems pretty weak. We are not trying to make Earth as hot as the Sun or anything like that. The only thing we want is to gather enough energy in a sufficiently small volume with oxygen and some fuel to light a fire; hence most of the energy for the fire still comes from the enthalpy of the combustion reaction.

Overall, I would think this is not impossible but probably very inefficient because of the minute fraction of power we receive from the Sun's light scattered by the Moon.

If you could fill the whole sky with moons you would not light a fire. It would be the same as looking up and seeing a wide expanse of bright shiny sand on a beach. What you can do with lenses and mirrors is no different than filling the sky with moons, so no: you cannot light a fire that way.