🗣️ Caught my cousin lying about me to the whole extended family, do not know how to react

Found out through my own mother, almost accidentally during a completely unrelated conversation, that my cousin had apparently been telling extended family members for the past several months that I had been deliberately avoiding a family gathering last year specifically because I supposedly looked down on that particular side of the family, which is genuinely not remotely true, I actually missed that specific event because of a work deadline that I had already explained clearly and directly to several relatives at the time it actually happened. Confronting her directly felt like the obvious first step, but she has a documented history within the family of denying things flatly even when there is fairly clear evidence pointing directly at her specific version of events being inaccurate, so I genuinely do not expect a straightforward admission if I actually bring this up with her face to face about it. What bothers me most, more than the actual specific lie itself, is realizing that several relatives have apparently been quietly forming an inaccurate opinion of me based entirely on her particular account of things, without anyone actually bothering to check directly with me first before simply accepting her version as accurate and moving forward with it unquestioned. My mother suggested I simply let it go, reasoning that clearing it up directly might create more visible drama than the actual original lie itself ever caused in the first place, which is probably practical advice, but it also means letting a genuinely false narrative about my own character just quietly stand uncorrected among people whose opinion of me does actually matter to me on some real level. I am currently leaning toward calmly correcting the specific record directly with the two or three relatives who seem to have actually believed her version most strongly, without turning it into any kind of larger confrontation with my cousin herself directly about the underlying reasons behind it all. Has anyone dealt with discovering a relative had been actively misrepresenting them to the wider family like this, and did directly correcting the specific record actually help restore things, or did it end up making the whole underlying situation feel worse than just quietly letting it go instead?