I think your argument is logically valid--that is, IF the premises were true, then the conclusion would be true. And I don't think it commits any formal or informal fallacies (except perhaps equivocation in the sense I'll explain shortly).
The problem is that it is unsound, because it has at least one false premise; hence the conclusion is not "made true" by the premises. Premise 3 is false. The government does not tell us not to kill no matter what. As you point out, it tells us not to break specific laws against specific types of killing. Typically, citizens are not breaking the law (and are morally justified) in killing in self-defense or to protect others from an immediate and deadly threat. And (legal) killing in war and use of the death penalty (where it is legal) are also not forms of killing the government tells us not to commit.
Now, we may have reasons to think that some or even all killing in war is morally problematic and even more reasons to think the death penalty is morally wrong. And we have greatly narrowed the scope of such legalized killings over time (in the U.S. and even more so, in other industrialized nations, most of which, for instance, have made the death penalty illegal). And we may believe that it is hypocritical to say some killing is OK but not others (though almost no one, perhaps Jesus excluded, suggests that you cannot kill, if necessary, in self-defense). But I don't think that the government is "saying one thing but doing another" in these cases, because the government, just like most of us, does not treat all killings as the same thing (hence the equivocation in the use of "killing").