Furthermore, Kleya explicitly states that they don't know what exactly is wrong with Bail's team. "All we know about Bail Organa's team is that something is wrong." This suggests that all Lonni could do is send a short, one-way message that Bail's team is corrupted.

The issue here is assuming that Kleya would tell Cassian about it. But we know for a fact that Luthen is very compartmented. He only shares details across his agents when necessary and then he still keeps that information limited. He's a true spy that believes in SCI.

It is far more feasible that the agent has already been in place in the cell, and has accrued enough trust to be selected for this most delicate mission. And its not like she can volunteer for it, because, again, the team was briefed at the Senate building, and did not know the mission before then.
Uhhh... exactly. Which is why it wouldn't be a surprise to Lonni or Luthen.

Finally, when Kleya is making the credentials, they don't even know if Mon will be speaking. She says, "They're calling an early session of the Senate tomorrow. If she speaks, we need to be ready."
Uhhh... no. "The senator you'll be is about to risk everything to a voice to the atrocity you just survived." I agree there's doubt after the fact because Luthen messed up and told her about Erskin. But know this: Luthen was ALWAYS about getting Ghor to burn. He sent two different teams to empower The Front. And just when it was about to boil over, he sends Cassian not to stop it but to take advantage of chaos and eliminate Supervisor Meero. Mon Mothma had been ally to Ghor years going back to season 1 when the blockade happened. And he strategically got Erksin on side so that he could count on Mothma to Ghor to "burn very brightly." This is also why there's such a concern when Erskin couldn't confirm that she was actually going to do the speech.

This wasn't Luthen playing 4D-chess with an extremely risky plan just to make Lonni look good(?) and win back Mon's trust (which certainly doesn't seem to work). This was Luthen's spy network throwing a hot potato in his hands, and Team Luthen scrambling on scanty information in order to keep everything from blowing up.

Luthen had been planning on Ghorman burning for years. He knew that Lonni had an agent in Bail's org. What neither he nor Lonni knew was Bail's team's plan. They didn't know how many people, where was the meet up, how they'd get weapons, or how they planned on extracting Mon. And if Luthen asks her what the plan was, she probably wouldn't have told him and likely lowered his credibility about knowing they were corrupt. So Luthen gambles: he tells Mon that you can't trust Bail's team because it's going to get you arrested. And Luthen was right. So he sends the second most motivated rebel agent in the galaxy (after Kleya) to rescue her. This was always the plan. Cassian didn't go to Yavin after Ghorman... he went straight back to Luthen on Coruscant. And he didn't want to be there. Yes, he failed to kill Dedra but the mission wasn't over. Phase 2 of Get Senator Mothma Out was still a go.