I'm using PHPStorm and trying to figure out how to make it stop squawking when a __construct() has @return void in its PHPDocs...
According the PHPDocs, void is valid, or the @return may be omitted. That being said, is there a way to fix this or is it a bug?

See this document for how to suppress the warning for a statement. You can hover the portion of code and select the option from the drop-down. You will have to do this for each affected constructor though.
I don't see an Inspection Tool option to suppress this warning in a way that won't expose other issues.