It's my personal choice
Choices affecting only the chooser are personal. The disagreement is over whether a third party is involved: and that is the question the phrase skips over.
What I eat is about as personal as decisions get, bound up with my body, my health, my culture, my family's traditions and my budget, and I don't need anyone's permission or approval to make it. People reasonably disagree about the moral status of animals, and when a question is that contested, the decent response is to let each person settle it by their own conscience rather than have one side's view pressed on everyone else's plate. I'm not telling vegans what to eat, and I expect the same respect in return.
The principle is one almost everyone accepts: what you do to yourself is your business, and what you do to others is not. Nobody is disputing it.
The dispute is over which category this falls into. Calling the choice personal presumes that the animal's interests do not register, which is the precise point under discussion. Stated as a principle rather than a conclusion, it is unobjectionable; deployed to end the conversation, it has assumed the answer. Compare 'what I do in my own home is my business': true in general, and not a defence of everything that might happen there.
- Don't force your views on me
Different wording, same argument. The response above answers all of them.
Assuming the conclusion somewhere among the premises, so the argument never supplies independent support. The formal name is begging the question, or petitio principii.
How it applies here: Describing the choice as personal presumes that no one else's interests are at stake, which is exactly what is in dispute. The description does the work the argument was supposed to do.
Applying a principle generally while claiming an exemption for one's own case, without giving a relevant reason for the exception.
How it applies here: The principle that personal freedom ends where it harms another is accepted generally. The exemption claimed here is asserted rather than defended.