Using those definitions no moral system EVER could be internally consistent as there is no tautology to start with. You'd have to start from an assumption, as you do in everything else. Since you're starting from assumptions anyway you can start from any assumption and remain as internally consistent as any other system.
As I said, you need to start from an assumption, but you can still compare those assumptions and the ways one draws conclusions from them.
If Billy claims that his core axioms are that getting lots of cake is good and that Johnny should be punched in the face, we know that it's not true and that his claim would instantly collapse under scrutiny. He doesn't get to claim any arbitrary silliness as his core axioms (whether he's knowingly lying or just kinda stupid) simply because we don't currently have the technology to probe his mind and prove that they aren't. His axioms still need to be something that he cannot eliminate, reduce or generalize any further using reason.
If Billy revises his core axioms a bit and now states them to be that pleasure is good and that inflicting pain is only okay as karmic retribution (so punching Johnny is okay because Johnny punched him first, yesterday), then great, Billy is now being a bit
more rational and consistent. He's still being pretty dumb, but at least it's an improvement.
My moral system, if you can call it that, does not offer any way to solve disputes between actors who actually hold different core axioms. If I run into a machine, alien or animal which truly doesn't share my axiom of pain being bad and starts dissecting me, then they aren't being immoral or internally inconsistent and there is no way for me to persuade them otherwise. However, if it was you who was dissecting me, you would be, because you
do share my axiom of pain being bad. I wouldn't need to convince you to act according to my axioms, only your own.
So, to put it differently: sure, our axioms of course cannot be logically justified. That's fine, I don't mind. However, my claim is that due to our brains' similarity, in fact me, you, Hitler, the aztecs, Gandhi and the guy from Daesh all have the same underlying axioms if we all just use reason to chip away all the non-axiomatic cruft. And that's why I say that some of those guys are
more wrong than me; I'm not comparing our axioms because they are the same, I'm comparing whether and how consistently we act according to them. If I believed that their axioms were actually different than mine, I couldn't say that.