I was chatting with a Khajit outside a city one day and they were explaining that they were not allowed inside the cities.

I have not tried to play as a Khajit and can only surmise that they game "allows" you into cities for the sake of the Quests that require you to visit thiem.

But if you were roleplaying as a Khajit... technically you shouldn't be going into any city should you?

When you play as a Khajit how do you reconcile going into cities... or do you just avoid them?

Kitty don't give a [censored] to be honest.
I prefer to use a "I'm a Khajjit, and I'm in your city, deal with it." sort of approach.

I role a Khajiit thief that I dont allow in the city. I use FT to enter the thieves guild and pretend I found a back way in there. I use them for the fence only. I will also hide myself in a hood ( I know my tail sticks out) and not interact with the people inside the cities and only go in at night to break into houses and such.. Other than that, I stay out of the cities as it does not make sense for me to be int he cities.

Think about J'Zargo. He recently arrived in Winterhold and I don't think anyone would deny him access to other cities - there's a difference between one scholar and a whole group of Khajiit who, according to the Nords, are likely to cause trouble and engage in illegal activities. A single Khajiit, especially after you're recognized as the Dragonborn, shouldn't be a problem.

If you don't want to start the MQ, you can always roleplay that you found an alternative way into the city. There are even mods on Nexus that add secret entrances through sewers and such.