I am looking to rent a car at Larnaca Airport that allows me to cross the border to TRNC (North Cyprus).
Cross Border Rentals to Northern Cyprus are not allowed.
Should you Cross the Border all insurances lose their validity. You may see stories on the Internet where people buy temporary insurance at the border. These are also not valid.
This is the situation for all rental car companies.
Ok, are you only renting in south center or do you have an office in TRNC and can maybe get the car at the border. I could get there by taxi from Larnaca.
Make a search the website will show you who/what is available but it seems that there are no rental car locations in North Cyprus.
Be sure to stick to office hours at non-airport locations