Why did Kido want the extradiction order?

Kido is a pretty smart calculating person and I think part of his intent was to see how much of an asset Julianna Crane is to the Reicht.

But as John Smith quickly deduces when they meet, that was not Kido's whole intent.

The scene leaves us with something unspoken to the audience between the men with only them saying,

JOHN SMITH: Tell me something, Chief Inspector. Soldier to soldier. You must have known before you came here I was going to refuse your extradition request. Why are you really here?

INSPECTOR KIDO: Sir, if I may ask...

JOHN SMITH: Listen to me, Erich. I want you to erase all record of that meeting from the log. Understand? He was never here.

However, as the season progresses with other scenes between them fully revealed in the season two finale, and really, even stemming back to episode 2.01 when Kido confronts Trade Minster Tagomi after the meeting with General Onoda about the building of the bomb, Kido comes to understand that Tagomi's role in putting the schematics in the Japanese science official's hand was about being equal with the Nazi's and ultimately his intent about avoiding war simply by implying they had the technology, backfired.

INSPECTOR KIDO: Trade Minister. I am all but certain you have passed those plans to the Science Minister with that Nazi colonel.

TRADE MINISTER TAGOMI: Are you here to arrest me, Chief Inspector?

INSPECTOR KIDO: Trade Minister, I know that this is what you wish, and yet you seem... troubled.

TRADE MINISTER TAGOMI: I wish for parity with the Nazis, to balance the power, to maintain the peace.

INSPECTOR KIDO: You heard the general. War is inevitable.

TRADE MINISTER TAGOMI: Possibly. But with a weapon like this... how can such a war be won?

Tagomi later then gives Kido a film featuring a massive hydrogen bomb that he retained from his time in the alternate San Francisco. Kido chooses to pass this onto John Smith so that Reicht, now under new command from Acting Chancellor Martin Heusmann, would no longer seek war with the Japanese, fearing a bomb *seemingly already tested that was 4 times as deadly...

MAN ON THE RADIO ...and the Reich forces would immediately stand down. At the 11th hour, war was averted by the shock revelation that Chancellor Heusmann had fabricated evidence against the Japanese. The brave sons of the Reich who had been deployed to the front lines, were stunned to learn that Chancellor Heusmann's betrayal was exposed by one of our own leaders... the GNR's Head of Intelligence, Obergruppenführer Smith. In a historic moment, viewed by tens of millions, Obergruppenführer became the first American to receive the Iron Guard salute in Berlin.

It's revealed in the season two finale that Kido and John Smith have been working together in general to avoid war, as Smith can use the film along with evidence of the Heusmann-Heydrich conspiracy of the assassination attempt on the Japanese Crown Prince to escalate war through a confession of a defective Nazi into the Pacific States.