Uploading a file asynchronously
Your proposed changes will have marginal impact.  The way IIS works, IHttpHandlers are not invoked until the entire request body has been transferred.  That is, the entirety of the file data is already available by the time your first line of code runs.  That means your stream copying is just moving bytes around in memory, and while that is not instantaneous it's probably not slow enough that it's worth making it asynchronous.

I think you already know this, but you should make the Postgres call asynchronous if that's an option.  With the current call, you have a thread taken up waiting for a remote process to do its work.  With an asynchronous call, that thread would be available to do real work.  Of course, that only makes a difference if there is an incoming request that's waiting for a thread.