It’s up to the DM, as everyone is saying
It’s a new weapon, it could be improvised, simple, martial, or none of the above; there aren’t really hard-and-fast rules about what falls in each category. You probably want to consider the nature of the weapon, both narratively (how hard would this be to use?) and mechanically (how much damage is this doing? what other properties does it have?), but deciding how those factors work out into the decision here is a subjective call that the game leaves to the DM.
But that answer isn’t super helpful
It’s good to know there isn’t any official “do this” answer—if there was, you should probably use it—but knowing that, by itself, doesn’t solve your problem of figuring out where to put this weapon. Saying “it’s up to the DM” does the reader no good, if, say, they are the DM, and looking for advice.
Which is why most answers also give opinions on what it should be. To not do so would, in my opinion, leave an answer incomplete.
So what should it be? (TL;DR: Simple)
Ultimately, the choice here is “who should have proficiency?” or maybe “how hard should it be to get proficiency?” Simple weapons are easiest (almost every class gets that proficiency for free), martial weapons a bit harder (only some classes get that proficiency for free), improvised weapons considerably harder (generally requiring a feat), and an answer of “none of the above” implies proficiency is impossible.
Simple or martial are the most appropriate choices
To me, “who should have proficiency?” is very, very close to asking “who should ever consider using this?” Attacking without proficiency isn’t impossible, of course, but the relative accuracy of attacks with proficiency means there’s almost never any reason to choose a non-proficient weapon. Maybe the claws of the umber hulk that Thomas Markov mentions have unique magical properties that justify it, but for anything you could just as easily replace with something you do have proficiency in, there’s not much contest.
So that leaves “none of the above” out as a reasonable answer. At that point, the DM is basically deciding you can’t really turn it into a functional weapon. And if that were the case, improvised would be far more appropriate.
But if we’re saying you actually manage to fashion a functional weapon here, it should be simple or martial. If it’s simple, its damage and properties should be worse; if it’s martial, its damage and properties should be better. But since you’re designing a weapon from scratch, you have a choice which way to start: you could start with stats and assign category accordingly, or you could start with a category and assign stats accordingly.
Historical precedent: Simple
D&D has had clawed gauntlets before. The “v.3.5 revised edition” of the game has both spiked gauntlets and punch daggers as core weapons—and they are both simple. (Weapon stats are different enough between 3.5e and 5e that the other 3.5e attributes of these weapons probably aren’t too meaningful for 5e.)
This also matches your own intuition, and the precedent set by stuff like the 5e dagger and sickle weapons, which seem most “clawlike” to me. Also, ultimately, the claws of a natural creature lashed to a gauntlet just... wouldn’t be that great a weapon, not next to steel swords and the like. So simple-weapon-like stats are probably most appropriate.