You could rub it off the tyre with some... Something you will find suitable. It's ok to do. But they're checking tyres every time IMHO and they would find out either way that you did something to that specific set of tyres.
The best way is to buy used tyres.
BTW. When you rent a car, drive careful on the track, it's even worse scenario if you crashed your own car.
Even if you get hit from behind by some noob, you will always loose, no matter what.
Does the track have some "official" cars to rent for trackdays maybe? You're saying you have low budget, but believe me, any unexpected situation happens on track - it will kill you in terms of finance when you're doing it this way.