The Corridors are intended to survive orbital bombardment. We're talking Castle Brian level construction. Sure your mechs could break them eventually, but not quickly enough to avoid discussing the fact with angry elementals.
Its more of a twist for a scenerio. And in the scenario the hall of the Khans would not actually be close, unless you're playing a "Attack the Khans!" type of scenerio where its a full on out of the sky invasion. In which case, all that underground fortress stuff was designed to deal with that. Also you would have to worry about the mech elevators then.
Your counter tactics... First you have to know where the entrances are. They are hidden until exposed. In the AU I originally designed the Obsidian Idol in, the Hall of Khans, and Svoboda Zemyla were the subject of a full on attack. One of the results is the Ebon Keshik fortified the living hell out of the entire area. Coincidentally, the canon story pretty much had the same thing, including orbital bombardment happen. No Idea what happened to the ebon Keshik from that point on.
Sending your own elementals in. Legit. But you're also asking to go up against elite units who are less interested in honor then in your destruction using top of the line equipment. In tunnels. That they know. Including where the murder-holes, booby traps, and secret passages are. Actually, that would be a fun scenerio in itself. Though more of a dungeon crawl then battle type of thing.
Cutting out mechs. Well for one, the Ebon Keshik is mostly made of elementals. They're a defensive police force. My version went through bad times and augmented their forces for more dakka...
In any case. If you're planning an actual trial, then the Ebon Keshik might declare they're defending with just elementals. If you're staging an actual assault, or decide to bring a dishonorable amount of firepower to the trial, then its time to break out the big guns.