Awesome idea, but perhaps the wrong time period and province for that build. It would have worked great in Morrowind and Oblivion, not so much in Skyrim for the reasons you and others state.
Khajiit away from Elsweyr are often Assassins for the Thalmor. I believe this is the main reason Khajiit are banned from cities; just go to see the Bosmer in the Grey Quarter, and then do his quest. I believe even the loading screen hints/historical notes mention this. I can't say anymore wthout delving into spoilers. Generally its just not worth the risks as far as the Jarls and Guard are concerned. Aside from moon sugar/skooma and other unsavory things, their perchant for thievery, and the racial and political tensions... if you let a caravan of khajiit into your city, you might just be allowing an assassin or spy free access to their target. Its a very cold-war stuation. I like the theme very much personally, and as above, just wish my Altmer was treated with more racism.
For the player, its a bit of a cop-out but the game probobly assumes that you go to Whiterun and get in only to report Helgen, and then once you are Dovahkiin no one is goign to dney you entrance to a city. You are a special case. Even if you are building exactly what they are afraid of.

Otherwise you can RP and stay out of the cities by choice, or go to Riften where you can RP that the bribe to get in was just because you are Khajiit, or just come up with something else. I don't play beast races, but with the tensions, Skyrim is the first I have been tempted to. They never interested me in Daggerfall, Morrowind, or Oblivion beyond their lore and being creatures I could meet.