The claim here is that eating meat is OK if you're a libertarian and that right must be exercised
I will assume that one means simply classical libertarianism, which is founded upon the non-aggression principle. It should nearly go without saying that eating meat requires express and intentional aggression towards a sentient being following a life of imprisonment, and therefore is amongst the most anti-libertarian ideas that exist. Any arbitrary classification that limits this non-aggression principle is not a libertarian principle, whether it's limiting it to humans, your race, or your family.
Most people have an idea that moral obligations belong to individuals based on intelligence. But why we can't treat infants or mentally-handicapped people - with the same intelligence as animals - like we treat animals? If you push a libertarian on this (e.g. Stefan Molyneux), the response is that somehow the rights of "being a human" are granted to the less-intelligent human. So even though this human - as an individual - shouldn't have the right to be thrown into a shredder alive, as a matter of the fact that they are a "human" has the same right as all humans, but this isn't extended to animals because they are not "human". This position is called "Speciesism". But imagine you are considering the ethics of some transaction between you and another individual. Speciesism is the position that the existence of other (similar) copies of that individual exists on the other side of the planet from you - that you draw a circle around and label with a common label - changes the nature of this transaction. That is, you are assigning moral obligations towards an individual not based on your moral obligations towards that individual, but rather towards the entire class. This is heavily anti-libertarian ideology. All I can say is that this line of morality-by-class and class-warfare thinking would make Marx proud.
In addition, libertarianism precludes many arguments that carnists would like to make. For instance, no health argument can be made since libertarianism is fundamentally not a consequence-based ethical system. You also can't make any sort of "hunting is good for the environment or for some species"-type argument since, again, consequences are not a defense for objectionable action. If your ideology is based on rights then to argue (poorly) for consequentialism is not an argument that you can or should make.
Well, truth is, it's hard to come up with a definition of political libertarianism because it's been co-opted by people who support rather anti-libertarian ideas (and in some cases are blatant authoritarians). So it's difficult to suss out what that means or if the crushing cognitive dissonance of these internal inconsistencies would bother someone who is all about "freedom" except in the case of - for instance - recreational marijuana, abortion, freedom from religious oppression, freedom from military imperialism, expanding civil rights and suffrage, ending corporate subsidies, and promoting truly free markets (not to name any names or anything). It doesn't bother people to hide behind the "libertarian" label and against all of those things, so why should it bother them if they add "not abusing sentient beings" to that list?
But I think it's fair to say that any political ideology that is aimed at some sort of well-being or rights for those within the system must take into account the well-being and rights of the animals, and anything else is probably just one of the other E-series of bad arguments on this site... with just a clean wrapper of "this is my political ideology".