The Lex Talionis was actually restrictive in it's nature. The remarkable thing about the law was that it set a maximum punishment, equal to the damage inflicted. In contrast, prior laws had said the equivalent of "a life for an eye." It was quite remarkable that limitations were being placed on revenge. This was not an imperative, but a restriction on the one demanding vengeance.
To answer you question, no it did not demand that an equal punishment be meted out if the victim did not desire the punishment be inflicted. Instead, it was saying, the victim may not demand any more than this.
The case law cited draws it out:
20 “Anyone who beats their male or female slave with a rod must be punished if the slave dies as a direct result, 21 but they are not to be punished if the slave recovers after a day or two, since the slave is their property.
22 “If people are fighting and hit a pregnant woman and she gives birth prematurely[e] but there is no serious injury, the offender must be fined whatever the woman’s husband demands and the court allows. 23 But if there is serious injury, you are to take life for life, 24 eye for eye, tooth for tooth, hand for hand, foot for foot, 25 burn for burn, wound for wound, bruise for bruise.
That women were property, for instance, would have been taken for granted. That a man might be put to death for killing someone's property is rather progressive for it's day.
Note also, it is the court here that is in charge- not the aggrieved party. If the victim did not wish the court to act, the court was obliged to do so. If the husband makes no demand, there is no case.
Indeed, the concept of judicial mercy itself was rather unique, and was developed over the course of the OT. The prophets over time called out for mercy. To hear the God of the Bible saying things like, "Vengenance is mine (I.e not yours!) says the Lord" is remarkable progression.
Additionally, unlike, say, even the Code of Hammurabi, Leviticus is clear that the principle is to apply regardless of social class. In contrast, the Hammurabi Code, while having the same principle, was clear that this didn't apply, for example, if a master plucked out his slave's eye. (The slave might be freed, but there was no such thing as equal compensation)