im looking for a chapter that is basically my own loyalist alpha legion.
Okay, here we go then. So what you want is a Chapter based on the Alpha Legion as they existed during their Loyalist days? In that case, I would recommend toning down on the 'unknown' quality you currently have in this IA. The Alpha Legion was certainly capable of operating behind the scenes, but it is notable that while they were still Loyal they were open about their involvement or presence. Imperial Commanders might not have had any idea what they were doing, but they did know they were present and they did know it was the Alpha Legion's handiwork. Therefore, one would expect the Inquisition to be fully aware of this Chapter's identity and the aftermath of their battles, if not the actual methods involved or the exact details.
It might also be notable that the Alpha Legion wasn't just a Legion of covert operatives. They were effective as such, and they went full hog in that direction when they went traitor, but they were also known for being something of a showboat. They'd spend an inordinate amount of time preparing, allowing unnecessary Imperial casualties, simply so that their first strike was the killing blow, and that it comes off as spectacularly as possible. Their covert actions was simply one tool in the shed for setting their ducks all in a row for the true battle, within which they were unleashed with gusto. If you work this kind of grandstanding behavior into this Chapter, it might go far into making them a bit more well-rounded and interesting as a Chapter. It's what makes the Alpha Legion itself so interesting.
However, I have to say that if this Chapter is meant to be based on the Alpha Legion, I am seeing little of the flaws such inspiration should bring. I'm not seeing any of the self-destructiveness that the Alpha Legion's secretive, clandestine nature leads to, which makes them seem, to my eyes, like a "new and improved" Alpha Legion-style Chapter. But there's also the fact that the Alpha Legion was able to operate as it did because it was a Legion, led by one of the Emperor's sons. They were top dogs in the Great Crusade, and the Imperium at that time was the Great Crusade. This level of power was stripped from the Legions afterwards, though a small number of Chapters still enjoy something close to it, like the Ultramarines, Dark Angels, Space Wolves and a few others. Since your Chapter simply isn't among that number (unless we are making an alternate 40k universe where this Chapter of yours is equal to these kinds of Chapters), the Inquisition, who would seem to have an antagonistic view of this Chapter, is the more powerful institution. A single Inquisitor can sign the death warrant of a Chapter, and for less cause than this Chapter seems to be giving. This isn't necessarily a bad thing, but it's something that we should probably be able to see. How is this Chapter handling the Inquisitorial threat, a threat that is more powerful and more clandestine than itself? Is it on the road to becoming a renegade, like the Sons of Malice? Or does the Chapter survive by handing the reigns over to the Inquisition, like other Chapters, such as the Red Hunters, have? Whichever way you go, we just need to see it, or at least witness how the process begins if the climax is meant to be post-'present day.' Otherwise, it might make it feel like there's too much going on, without what we might feel as adequate explanation for why it is.
im looking for help on how to make it work while still leaving room for people to wonder about their actual origins and such.
There's a general writing tip I remember from way back when that went along the lines of "Show, don't tell." At the moment, the 'unknown' nature of your Chapter is being told to us. It might be best served if it was shown. You could, for instance, let us know what the Chapter's name is. Let the Inquisition know. But, in the writing of this article, show us how this identity is kept under wraps, denied or perhaps even faulty information. Do this not by saying that it is, but by showing us something that implies that it is. Maybe the supposed progenitor denies any connection to the Chapter. Maybe an investigative Inquisitor finds records of a Chapter bearing this name as being destroyed. Were the records faulty, and this was simply a defeat the Chapter survived? Was this Chapter's identity simply handed to a newly Founded Chapter following its destruction, or was its identity usurped by this other Chapter?
they are focused on small ops.
I would recommend adding in more than this, lest they be mistaken as 'one trick ponies.' The Alpha Legion itself, for instance, was far more than just a small ops Legion. The thing that made them effective, as an example, was their tendency for "multi-vector attacks," which was Forge World's fancy way of saying that they liked to set the enemy up and strike with everything they had from every potential angle all at the same time. Overwhelming the enemy, who cannot possibly defend against every attack simultaneously.
the inspiration is The Serpent Beneath from The Primarchs.
This story shined a spotlight on the self-destructive nature of the Alpha Legion. How it was a threat to itself, by its own secretive tendencies. I would really like to see that in your Chapter. I think it would go far to flesh them out and make them more believable as something that exists in this grimdark setting.