I don't know what the grammatical reason for it is, but I can tell you that no, "Min far äter kött inte" is not a correct sentence, though the other two are both acceptable. My best guess for the difference between those two is the second carries the implication of not loving that person specifically ("I don't love you [but I love him]") or possibly just as a cold statement of fact. The first is more "emotional" so to speak, kind of like negating the phrase of "I love you" as a whole, rather than just "subject verb negation object". That may be the difference, since "my father eats meat" isn't exactly a particularly common phrase that gets thrown around a lot. That's just my guess though, so I may be completely off.
Tl;dr: No, you cannot say "Min far äter kött inte".
Sorry, I don't get it: is or isn't "inte" in a clausal adverbial position in "jag älskar dig inte"? And if not, why can it be outwith it?
Doesn't matter why, the important thing is to know that it's the pronoun which makes the difference. Tack så mycket för ditt svar!