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.)

I can find no Sage Advice regarding the matter, but the rules about unseen attackers starts with this:

Combatants often try to escape their foes' notice by hiding, casting the invisibility spell ...(PHB 195)

This seems to suggest what I believe every table uses: If you are invisible, you attack someone who cannot see you with advantage, even if doing so breaks the invisibility.

Regarding nocking an arrow or swinging back for a punch: this would be a Ready action and not Attack or Cast a Spell, and thus would not end the spell invisibility.

If you cast a spell with a longer casting time

you must spend your action each turn casting the spell (PHB 202)

thus you reveal yourself on the first turn as you have taken a Cast a Spell action.