Want to upgrade ELK cluster from 5.6.x to 7.x w/o Kibana downtime

I'm working with a 12 node cluster, so upgrading every node is likely to take a while. Currently when there is a version mismatch between any two nodes in the cluster, Kibana refuses to load. Is there any way to work around this problem so that Kibana can be online while I do this upgrade?

You could set the upgraded nodes to a separate cluster and when they all finish upgrading you can point kibana to that cluster. But do keep in mind you settings for data replication and so on.