[doubleunplussed]

The rent isn't based on the sale price of the land, it's based on the value of the land. If land comes with a tax obligation, they're not the same thing anymore.

    sale_price_of_land = underlying_value_of_land - how_much_you_would_pay_as_a_one_off_to_avoid_paying_the_tax_forever

So no, you wouldn't update the tax rate to be a fixed percentage of the new sale price of the land after bringing the tax in. That's not the value of the land, it's just the sale price.

Instead, you would update the tax rate in order to ensure the sale price of unimproved land was zero. If the sale price is zero, that means the tax rate is correctly set so as to cancel out the land value as per the above equation.

It's weird having the value of the land plus tax obligation be exactly zero, which is one of the reasons many Georgists propose merely a 85% tax instead of a 100% one.