SQL Server 2012 mirror setup - how to reduce transaction log files
Direct answer to your question


  What would be the best course of action to reduce the log file sizes? 


In full recovery model only transaction log backup, perhaps frequent transaction log backup in your case, would truncate the logs and keep check on log file size. There is no other way to truncate logs. I strongly suggest you to read Why transaction log keeps growing and run out of space


  Should I go ahead and break the mirror, switch from Full -> Simple then back? Does anyone have experience with this kind of setup?


No please dont do that this will break the mirroring and you would have to re-establish it and it would be big hassle for you if you have got big database and mirror is located far from principal. Anyways if you do that and somehow configure mirroring after few days you would again face this issue because you actually did not solved it.