Running SQL-like queries over large schemaless JSON dataset in the cloud?
You want to write structured queries (have your cake) but have unstructured files (eat it too). 

Is there any way you can play with the files a bit and have a common schema between them, with some fields just empty? That would make it easier.

That being said, BigQuery is probably your least bad solution as you found. Amazon's new Redshift Spectrum is kind of similar but will also require a schema to be defined.