How do I lay "click system" laminate flooring across an opening?
First of all, I disagree with your orientation.  But, it is your choice, and I'll respect that.

You may want to read this: http://diy.blogoverflow.com/2012/09/installing-laminateengineered-wood-floating-floors/

Pay close attention to the pre-measuring, so that you avoid having a 1 inch board running along either wall.  

The idea is that you never want to rip a board to less than half it's width.  If you end up with a half - board gap along either wall, then you should really be using a 3/4 board gap along BOTH walls (That is, convert the final 1.5 board widths to two 3/4 board widths at start and finish, or whatever fraction works for your measurement.)

It's going to be particularly tricky in this case, because you have 4 different measurements to deal with (three rooms + opening).  However, it is an important step. It's going to be some tough geometry to strike the right compromises.  You might find it easier to work in Board Widths rather than inches.  Just be careful with the conversions!

Don't be afraid to notch the board so that it slips around the wall if required. Again, see the blog post for measuring tips.

Once you've figured out the measuring, proceed as The Evil Greebo suggests.  However, you're going to want to start against the wall that will leave you the smallest amount of floor to slide around to make it connect up.  This material has to be aligned exactly for it to snap together.  From your drawing (which may not be to scale) I'd start at the right side.

Your wiggle room is the width of the baseboards and quarter round you're going to use to finish the job.  You need a 1/4 inch gap from all the walls, and a typical baseboard + 1/4 round can cover up to 3/4 of an inch. (But do a recon at the Home Depot and actually measure this stuff so you know what errors you can cover up)