using North America Web/Internet settings
In a non-color managed application, the display of the document might be different on each display. But the file has no dependence on the display.
Photoshop assumes all untagged files are in the sRGB color space, but all Apple applications, from the Finder through Safari through iPhoto, throw Monitor RGB at untagged files (dumb!).
If Save for web doesn't give you the option to embed a profile in a PNG (haven't checked that yet), you can always use the "Embed Chosen Profile" Apple script included with your Mac.
"On the Mac, the Application frame groups all the workspace elements in a single, integrated window that lets you treat the application as a single unit. When you move or resize the Application frame or any of its elements, all the elements within it respond to each other so none overlap. Panels don’t disappear when you switch applications or when you accidentally click out of the application. If you work with two or more applications, you can position each application side by side on the screen or on multiple monitors. If you prefer the traditional, free-form user interface of the Mac, you can turn off the Application frame. In Adobe Illustrator®, for example, select Window > Application Frame to toggle it on or off." - Adobe Help "Using Photoshop CS4"
Once you have done that, move the whole block of the Application frame to the Display Tablet. You can then toggle between the monitor 1 and tablet monitor by switching on/off Application frame..