Advice on transaction log based reporting server
Is there a better solution for what I am attempting to accomplish where the database wouldn't need to refuse connections, or could queue connections until the backup has been applied?  

You need to write some manual code and change backups .Please see below article which we are also applying from Aaron bertrand   

http://sqlperformance.com/2014/10/sql-performance/readable-secondaries-on-a-budget