Splitting a mac desktop into smaller portions Background

I'm using a device called a DualHead2Go that lets me plug two external screens into an early gen Macbook Pro. This device basically presents the two screens to the mac as if they were one gigantic 3840x1080 screen.



The macbook's internal screen is on the right

The Problem

For the most part this setup works really well - the problem I have is that, as far as the Mac is concerned, the middle of this desktop is right in the break between displays. Which leads to situations like this:



Since this is my primary display, notifications and modal dialogs often come up right in the monitor split - not only is this ugly, it's a PITA to deal with.

Solutions

In order of preference:


Some application that would make the Mac work as if the "one display" were actually two. If this were a Linux box, I could do some trickery inside Xorg to present the one display as two.
Some application that would allow me to tile windows, taking the display into account


I've tried applications like Divvy before, the problem there is that they don't cleanly divide the desktop, leading to windows overhanging the split

A tiling window manager, or some other application that allows me to divide my desktop space into set pieces, including where dialogs and notifications pop up.


Are any of these viable?

Questions this is NOT a duplicate of:

Is there a way to Split one 42 Inch TV into Multiple Displays or Virtual Displays? (More than four years old, and the one answer to the question doesn't actually answer the question)

How applications behave with Matrox Dualhead2go? (More than two years old - the application situation might have changed)
Have you looked at PowerDesk for Mac? the description says it can do what you want. There's a DualHead2Go &amp; TripleHead2Go User Guide for Mac. PowerDesk for Mac can be downloaded for 10.6 thru 10.10. See Download info.

Okay, instead of preferences in order, from your comment they're all requirements.

Try a combo of PowerDesk for Mac, Cinch and SizeUp.