Is there a way to toggle to a web browser while in Keynote presenter view?

I teach at a number of research classes at a University and I prefer to use Keynote as a teaching tool. My teaching requires that I access websites regularly throughout the the class.

Here is what I'm currently doing:

1. Before the class starts, I open each website we'll be using during the class as window and assign the window to it's own desktop. So if I have four websites the students will be accessing during class, I'll assign those four websites to four distinct Mission Control desktops (I'm using the 10.8.4 OS X).

2. I run my presentiation from Keynote using the play key, but without access presentation mode.

3. When I need to access a website while teaching, I [ALT]-[TAB] out of Keynote, select the appropriate page from Mission Control, and demo that page.

4. When I'm done, I simply click the Keynote icon in the dock and resume my presentation.

This system works and I have no real complaints with it - except that I cannot take advantage of Keynote's presentation tools. I've tried any number of ways to toggle to a web browser while in presention mode without success. While in presentation mode, if I [ALT]-[TAB] out of the presentation, it simply stops the presentation and the web page doesn't appear on the projection display (because I'm not mirroring my displays) and the only way to get it to work is to re-mirror my displays.

I tried using the X command in Keynote, which is the toggle between displays, but no matter what I did, I cannot get the webpage to show up on the projection display.

Is there an easier way to do this? Bear in mind, I'm not merely showing webpages, but I'm actually interacting with them (performing research on these pages) so they have to be operational. Right now I'm assuming presentation mode is just not going to work for this kind of scenario.

Thanks for any suggestions.