Combat Trials and bidding would likely determine the order like Operation Revival did in canon but there could be a fair amount of operational control dictated by the ilKhan.
Some general thoughts on your Invading Clans -
Blood Spirit (BS) - Probably paired with Fire Mandrill and Snow Raven with an outside chance for Diamond Shark
Burrock (Bur) - I'd incorporate them into the 'Snake Alliance' which should include Star Adder, Steel Viper, and Cloud Cobra
Cloud Cobra (CC) - with the bloodname Steiner it would be interesting if they were in the Lyran Commonwealth
Diamond Shark (DS) - the reserve Clan before they could be again here, not in the forefront, just happy to expand their markets
Fire Mandrill (FM) - Kindraa Sainze going back to the Draconis Combine might be too good an angle to pass up despite their divided nature
Ghost Bear (GB) - I like where you are going with the three core Clans but why not have them in the middle and in Rasalhague again? Be a nice similar theme with them identifying with the natives and thinking about settling down there.
Hell's Horses (HH) - tough one here as they might get the shaft since their blood rival is running the show, can you say the Periphery flank, they might be extremely well suited here however
Ice Hellion (IH) - love to see them against the LCAF just tearing up their heavies with lights
Jade Falcon (JF) - love your core three idea but again would have them just the Lyran side of the Bears
Nova Cat (NC) - could the Bear ilKhan have the Cats along side the Jags again just to slow a rival

Smoke Jaguar (SJ) - love your core three idea but again would have them just the Draconis side of the Bears
Snow Raven (SR) - I'd pair them closely with the Bears, perhaps like the Sharks they could be a reserve, just happy to trail along
Star Adder (SA) - love the 'Snake Alliance', I'd also like to model the Klondike relations for the Bears and Adders here, tense and at odds with the Bears senstive to Adder military strategic advice
Steel Viper (SV) - love the 'Snake Alliance', perhaps like the Cats and the Jags, the Vipers go next to the Falcons in order to slow a rival
So my center would have the Ghost Bears in the center flanked by the Falcons and the Jaguars with the 'Snake Alliance' drawing the Lyrans and the weaker Spirit-Mandrill-Raven crew getting the Draconis Combine. Ravens you could have on the Periphery Draconis flank so they can connect to their Outworld canon theme plus it pushes the weaker Clans to the outside of the Combine. Nova Cat would flank the Jags. The only two I had to fiddle with were Hell's Horses and Ice Hellion (which I unfortunately couldn't place in the LC) in my first option.
DS
HH-Bur-CC-SA-SV-JF-GB-SJ-NC-IH-FM-BS-SR
Although I could pull a switch from Hell's Horses being assigned all the way to frontier duty to the Wolves Revival spot which does stick to the theme. Still doesn't move the Ice Hellions to the Lyran side or address the weaker Clans being in the Draconis Combine but hey....
DS
Bur-CC-SA-SV-JF-HH-GB-SJ-NC-IH-FM-BS-SR
What do you think??