Kuribayashi is a reckless nutcase, and that's the clear reason from this episode and the city defense one why she's still with the grunts and Itami is the decorated special operative that she desperately aspires to be.

But if you view that scene in terms of demonstration, it makes a lot more sense. You have Itami and the Japanese ambassador standing in front of the Emperor, his successor, several high ranking officials, and their first princess. She already knows what they can do, but a lot of what she's said to other people has been brushed aside as stories, until they see it for themselves at a villa party.

There were others there. Itami could have had Tomita back up Kuribayashi and finish the whole thing in seconds. Tomita only covers her, and never once fired.
Itami deliberately let her wade in alone, knowing she'd charge in with a bayonet. It was a demonstration in escalation of force.
- The JSDF send their "weakest" fighter, a woman.
- Their "weakest" fighter can solo a number of their soldiers with her own blade, not even using her terrifying primary weapon.
- When they use defensive combat, she fires through them in seconds.
- Itami then has Kuribayashi drop her weapon and beat the shit of a skilled fighter with her bare hands.

All while the Emperor looks on, seeing first hand the difference in their individual skill. Not just because Japan has super-weapons compared to them, but because they are simply better trained and superior fighters. A tiny JSDF woman can beat them with a blade, she can tear through their defenses, and even without a weapon she can beat some of their best.
That's why she starts with a bayonet. "No matter what we use you can't beat us."

The trick is of course that Kuribayashi isn't their weakest fighter, she's one of their nastiest (though poorly disciplined). But she looks like she would be the weakest to these people.