Nightblade. Sorry, but that is the best fit thematically, both lore-wise and gameplay.

EDIT: As a side note, I have a Khajiit Stamina Arcanist DD that I love for group PVE endgame. So if you are looking to min/max with an Arcanist, its a great call.

If you want to thematically design your khajiit character then you can just focus of the preffix/suffix within the lore. Anything would fit the approriate class such as -jo/ for templar, sorc or -dar/daro for nb, -do for DK? Etc...
There's no real lore reason why you wouldn't have a khajiit of any particular class.

For the lore side of the choice, I would suggest to think how your khajiit would embodied a class, for example :

- DK are tied to akaviris, and we see descendants of the akaviris in Elsweyr, wich is also a imperial province quite close ( at least for the northern part ) to Cyrodiil; also . So a khajiit dragonknight can be a sound choice.

- Sorcerer is a class with a quite extended identity in my opinion, wich combine mastering magic ( especially weather magic, most notably lightning ) and bending lesser daedras to their will / using 'deadric magic'; but you could extend that to mastery of some others magical domains, at least elementals one ( fire, ice ) ( I have no idea if there is something like an established classification of magic domains in the elder scrolls ). Anyway, even if khajiits are, I think, not really well known for their spellcasting abilities ( maybe the little khajiits ? ), I don't think they're unfit for being a sorcerer. And like every other races, kahjiits have some relationships with daedras ( both in actively worshipping some and in their myths ).

- Given that, arcanist are not a bad choice for a khajiit : some of them worship Hermaeus Mora, and there's even a secret temple to him in Reaper's March. Khajiits also have some impressive mythology and an interesting relationship to it, so a class wich is built upon not only knowledge but also stories, legends and such, would certainly fit a khajiit.

To conclude I would say two things : not all khajiit are thieves, it's an obvious choice but certainly not the only one; and mine is a necromancer, in part because of the quest in northern elsweyr where you face a khajiit lich ( or something like that, but definitely necromantic ), so I thought it would be cool to make a cat that love digging bones.

Be an azurah worshipping necromancer, a spirit welding healer purrrrhaps?

Lorewise there are Khajiiti monks, healers, knights, priests, mages, scouts, scholars, grave diggers, gardeners and much more. Not just thieves, assassins, beggars and shady merchants.

So, in my opinion, every class can fit the Khajiit. It is a matter of how creative and adaptive the player is.