Describe it differently
The crux of this seems to be "Wonder Woman's Invisible Jet is silly," which is an issue with the fiction the rules inspire. I agree, it's a pretty silly thing to have to imagine. In this case, though, let's ask ourselves the question: do I have to imagine it that way?
We're talking about a beast that blends with darkness and shadow, not a wizard's Invisibility spell or a Predator cloaking device. If the whole problem is the idea of a halfling awkwardly floating above the ground, legs akimbo — well, imagine something else.
Describe it as "I'm riding a smokey blob of shadow," or "I'm riding a weird beast-silhouette with edges that waver like dark fire," or "I'm riding a regular ol' dog but its coat blends so well that you can't figure out where the dogs ends and the shadows begin til it's in full motion jumping out at you like RAWR." I promise nothing important will break if you say that you can't literally see through the big magic darkness-blending shadow-dog to the rider's leg on the other side.
Mechanically, you don't need to change much: the dog is still "invisible" as far as the rules are concerned, and the rider isn't. Feel free to describe being able to sorta make out the dog's shape or presence while it's got a rider, since just seeing the rider is enough for enemies to know its location and take a swing at it even if it's invisible. Feel free to describe how the rider is partially obscured by the dog's fur or its aura or whatever, but you can see most of them pretty clearly (that doesn't require special mechanics any more than "this broad-brimmed hat I'm wearing casts a shadow over my face" requires special mechanics).
There may be some small points where the mechanical and fictional description diverge, but in my experience that'll happen with detailed rules for seeing things in D&D across editions anyway. Just invoking the detailed concealment or invisibility rules has a tendency to bring out goofy edge cases.
(Experience: that's how we've treated shadow-creature shadow-invisibility whenever it's come up across different editions of D&D and it works pretty fine. Not transparent, just impossible to isolate from their environment.)
Minor conjuration can be done about every 6 seconds
[...] you can use your action to conjure an inanimate object [...]
By default, PCs can perform 1 action per round and each round is approximately 6 seconds long. Thus, it would be possible for the 2nd level wizard to perform a minor conjuration about every 6 seconds by consuming their action. If the wizard obtains a way to gain additional actions per turn that they can use to perform minor conjuration, then the amount of times it can be done is multiplied accordingly.
There is no other limitation to how often they can perform it.
Minor conjuration is not a spell and does not interact at all with a spellbook
Minor conjuration is not a cantrip or any other type of spell; it is a class feature. Thus it does not get written into a spellbook nor does it cost spell slots. Losing your spellbook will not impair your ability to do this either. In fact, as written, only one thing is required to perform minor conjuration: an available action. Nothing else matters.
Best Answer
Absence of one light source does not affect the brightness of another
The brightness of a torch is not contingent on overlapping its light with some background light source, such as moonlight. This is evident from the fact that a torch works just as well underground in a pitch dark cavern (where there is certainly no moonlight) as in a moonlit (or non-moonlit) night on the surface.
Furthermore, moonlight is normally already considered darkness, as spelled out in the description of darkness in the Vision and Light section of the rules (emphasis added):
So there is generally no mechanical difference between a moonlit night and a moonless night anyway.
I can't find anything in the rules that explicitly spells out how lighting from multiple light sources combines, but the most logical rule I can think of is: the brightest light source prevails. That is, if a object is illuminated by both a bright light and a dim light, that object is brightly lit. If another object is illuminated by two sources of dim light, it is dimly lit.
However, your DM can make new rules
Despite the above, the DM can implement any rules they like. It's possible that the "nonmagical" darkness of this moonless night is just a different kind of magic. The Sage Advice Compendium explains:
Given this, it would not be unreasonable for the DM to decide that in the world they are building, the darkness of a night with no moonlight is not a mere absence of light, but in addition it has a slight supernatural edge to it that makes all light a little bit dimmer. This would be especially appropriate in a horror-themed campaign.