Is it permissible for me to enroll in a course at its old price as advertised on the platform, or must I pay the additional amount requested by the instructor outside the platform?

There is no disagreement among the scholars that a lease is not valid except with a known rent, and if it is agreed upon at the time of the contract, then the lessee is not obligated to pay anything more than that. If the hirer requests a price higher than what was previously announced at the time of the contract, then both parties have the right to negotiate until they agree on a known amount upon which the lease is concluded.