You're correct, he looks incredible and you're a brilliant artist.
It makes sense that Duke Specter isn't curious about Spike's past; earlier in the comic he expressed scorn because the Independent Privateers are not as glamorous as he expected, despite the fact that he is only alive and being humored because they're basically benign buoyant hobos.
Learning more would just continue to shatter the illusion that he has in his mind; the only problem is that Duke Specter is growing more fond of these weirdos than he had ever planned and his natural curiosity is starting to bloom towards people who arouse his interest.
Thank goodness he's too smart to catch feelings for a dirty pirate boy!
That would be waaaaay worse than finding out his fantasy is not 1:1 with reality.

In fairness they may not just be "Pirates" but "HIS pirates" but also it doesn't seem like they've all truly gotten to a place where, to Specter, they are more than his "Special Spring Break in another state friends that fill a role in his kidnapping pirate fantasy" yet. All his relationships are all in degrees of the Casual/Physical spectrums, and outside of having some crew mates that he favors more than others his attachment to everyone member isn't really deep or emotional yet, as if he is prepared for not only his fantasy to be broken if he get's too attatched to the crew as people rather than 'actors' in his game, but for some catastrophe to appear to physically end the play they are all in and he is not digging deeper in order to not have to feel a deep level of emotional loss should that happen.

If it was a otome game, Specter plays options passably, earning enough points to not get the bad ending but not playing perfectly enough to unlock all the love interest's backstories and special events, because once he does the game is over

That is an excellent way to phrase it, and I think that both Spike and Specter in this AU are hesitant to dig deeper for fear that they'll ruin the comfortable role that they both have right now.
If the duke was forced to go back to his former life right now, he'd be roughly back exactly where he was before, with the situation slightly worse because it's probably an open secret what he's actually been up to.
It's not like he's making any real effort to keep it a secret, or that the Empire can do squat about it right now.
Unless of course something significant were to change in the near future and throw everything into complete chaos.

Oh no they're both trapped in the YA loop of "Not wanting to ruin their friendship but have mutual interest in each other"-itis.
Inevitably something will have to explode and throw everything into Chaos tho. Preferably with a gut punch