Is it permissible, Islamically, to demand the penalty clause stipulated in the contract, which is the payment of one month's rent in advance, in the event that the tenant wishes to vacate before the agreed-upon term ends? And is swearing to not recall the details of a previous conversation with the tenant considered a sin?

If the penalty clause for the tenant's termination of the contract is the payment of an amount exceeding the rent for the remaining period, then he is only obligated to pay the rent for the remaining period. The penalty clause is stipulated in exchange for breaching an obligation. If the penalty clause is excessive by custom, then fairness and justice must be resorted to. However, if you stipulated that he pay a specific amount only, without the rent for the remaining period, and he agreed, then there is no harm in that. If you are not certain of your agreement to accept an amount less than what is due, you are not obligated to accept his offer, but forgiving him is preferable. And swearing a false oath is a "Ghamus" oath, which requires repentance.