Good news, we managed to do it.
We configured our ISS as a new ICP in our cluster. That way we could assign 2000 IP licenses to it. We shut down the MXe Server, isolated the ISS from the network, changed the MSL network info to use the MXe Server IP addresses, and restored the MXe Server database on the ISS. The licenses stayed on and all the users & devices appeared correctly. We did a clear hardware of the MXe Server on the AMC. We then connected the ISS to the network, did a couple of AMC sync from the ISS and the DLM, as well as a Network Element sync, and all is well. The phones came back from resiliency, we didn't have to intervene physically on them (fortunately, as they're spread out across the country and planet). From the POV of the cluster members it's as if the MXe Server was shut down and came back a bit later, with a new ARID. MiCC also working, LCS, LBG, etc. The only thing we didn't think about was SMDR so we lost a day of calls, not a big problem.
We just have to remove the cluster element/network element corresponding to the ISS before the restore, we're not doing it right away as we'll use it again to migrate the 3 other MXe Servers. We've restored that "before the restore" backup on another ISS and synced it with the AMC and cluster without trouble, our next migration is ready to go.