I rented a Civic Type R but i didnt want to risk it taking it on the track because on the Fuji Speedway website it says "If you are planning to drive a rental car or bike on a circuit, you must obtain permission from the owner". Not sure if they will actually check if you've got permission or not but I don't think regular car rental companies will allow you to take their car on a track and if you do it secretly you'll not be covered by insurance if anything happens.
https://www.fsw.tv/freeinfo/039913.html
(Use Google translate on Chrome, scroll down until the notes section, it's highlighted in yellow)
Thanks for the heads up. I'm thinking about just asking Toyota Rent-a-Car whether it's allowed. I would imagine this isn't a particularly uncommon scenario with their cars out of Shin-Fuji and it's not like the track experience is pushing it hard on the circuit. If they don't allow it, then I'll consider just using the track cars.