ReportServerTempDB in AlwaysOn Availability group
Microsoft describes it's support for Availability Groups with SSRS in BOL. In particular is the section titled "Report Server Databases and Availability Groups."  In my experience, having ReportServer and ReportServerTempDb be part of an AG works fine in most cases, but there is some peculiar behavior with subscriptions and failover--which is described in the BOL article.

Having independent ReportServerTempDb databases on each server, but not participating in the AG is not something I would try--but it would likely be problematic in it's own ways. It would also not be a supported configuration.

You should look at this answer to see why your transaction log is growing. I suspect it is due to an open transaction in the ReportServerTempDb. On a number of occasions, I have seen the CleanBatch process get hung, and create a scenario where it has an open transaction but is performing no work. The net result is that your transaction log gets huge. If you see the same behavior, then simply kill the spid with the open transaction, and things should return to normal.