Only the caravaners say they are not allowed in the cities, it isn't said or implied by anyone else.
I figure it means they aren't allowed to sell their wares in the cities, not that they are completely barred from entering.
This, only the caravaners speak of this.
The Jarls already have vendors in their holds, who [censored] and moan incessantly already about the Dragonborne flooding their markets with iron daggers. So outsiders, who lick their butts, strolll up to the main gate and want let in. These vendors will sell stuff, aquire septims from the townsfolk, and then mosey on down the road. That's another big issue. Those septims go into the caravaners pockets, where the Jarl can't tax it.
But most importantly, the OP is asking gamesas to impletement barriers to the player simply because of a race choice. Arbitrary barriers from seemingly inconsequential, at the time, choices is the hallmark of
poor game design.
Why are Khajiiti player characters allowed into the cities without any hassle? Really simple answer to that question, because that Khajiiti is being controlled by the player.