Was it out of character?

[the phantom]

attempt to kill the one in front and thus become a legitimate target for the one in back

Ah, but if he is a killer and has already killed in the past why is it necessary for him to attempt yet another killing before he is considered a "legitimate target"? Is he not fair game already? Why let him kill another good person when it is not necessary?

This is not to say Frodo should stand up and wave and say "I've got your Ring, now stand down!" to Sauron

Ha ha!

he, in a sense, becomes like the Dark Lord

I don't think that's what happened.

What was MoS doing? He was insulting everyone, attempting to anger and sadden everyone, and trying to get them to slip up and mention Frodo's quest. If Gandalf had not silenced Merry and Pippin one of them might have said "Frodo! Oh no! That means Sauron has the Ring, now!"

If MoS heard that, he'd immediately run back through the gate and tell the Nazgul and they'd tell Sauron and then it would be obvious to him what was happening and the Nazgul would be sent to Mt Doom and they'd get the Ring.

Obviously, Aragorn couldn't let that happen. He also couldn't afford to sit there all day being insulted. He had to start the battle as quickly as possible to make sure Sauron was focusing on them. It seems to me that killing MoS immediately was the smart thing to do. It hastened the battle and ensured that no word of Frodo's errand was blabbed.