I agree with Tassa. That's not a thing, there is no process.
A series of things would have to click into place that seem almost impossible. Basically, every other Kindraa would have to decide to accept the newcomers and be willing to support that decision. Given how little the Kindraa actually agree on anything, let alone a question/issue of this magnitude, that seems unrealistic.
As part of that, to be successful, I'd imagine this new Kindraa would have to bring a lot to the table. So they'd have their own equipment, their own enclaves etc. Because the other Kindraa aren't going to give them that stuff. Why would they?
More than that, they'd probably have to shower all the other Kindraa with gifts and things in order to curry their favor and support to get this going. It would truly have to be a deal that the Mandrills can't refuse. Here's a free cluster's worth of equipment and a pre-fabricated BattleMech factory.
But that same process would also carry a lot of suspicion that this is some kind of trick by another Clan intended to infiltrate the Mandrills and destroy them from the inside. A kind of doom loop that I don't see how you can break out of.
On top of that, outside of the Mandrills which treat the Kindraa as Clan-level entities for purposes of ownership, Bloodname Houses don't really own their own stuff. Warrior equipment, enclaves, sibkos, genetic legacies. It's all owned by the Clan, not the Bloodname House. So unless the originating Clan basically orchestrated this plan, it likely doesn't happen. If the originating Clan does support this attempt to create a Kindraa, that adds to the doom loop logic that this is a trap of some trap designed to infiltrate the Mandrills.
Could someone have perhaps coughed up an outlandish version of this that worked during the Wars of Reaving era? The survivors of a Clan, that happen to be of a particular Bloodname House, flee to the Mandrills before they starve to death? Perhaps, but even then I'd expect the Mandrills to just absorb that into one or more existing Kindraa, not stand up a new one.
Sorry, I don't see a path forward on this.