Can I float click laminate diagonally without transitions in my space?
Answer(s)

Is diagonal placement possible? - Yes

Can you do it yourself? - Yes - but make sure you have the right tools in advance.

Is there a recommended room to start in? - Given your floor plan - I'd start in the dining room, as it looks like the single longest continual diagonal line runs from there into that little side corner room thingy that's past "E" on your layout (nice job there btw).

For diagonal placement in particular, all of the resources I've read so far talked about "starting in the middle of the room" - consider the "longest continuous line possible" - start there.  Running on a diagonal, you have a few options and you don't have to start on the absolute longest but find a good long one to use as your starter.

Any specific pitfalls I should watch out for given my floor shape? Yes - getting your first line right will be critical.  If you're going for a 45 degree angle, get it perfect.  Measure 3-4 times, mark start and end points, snap solid chalk lines (use hairspray to "glue" down the chalk), and check and re-check.

Next, using your chalk line, line up the boards on your starter chalk line.  Lay them along the line but be prepared to shift them along the line towards one wall or the other, so that at either end your distance from last FULL board to wall is about the same.  

Initially your floor is going to want to shift around on you.  Once you have it where you want it, use a temporary means to keep it stable.  Take small blocks of straight wood and line them up against the edges of the starter strip and screw them down in place.  This will "lock in" the floor temporarily - and give you anchors you can move as you progress.  Once the floor becomes so large and heavy that it doesn't shift around on you, you can stop with the blocks - but given the length you're dealing with, that'll probably only be after you've got a good chunk of the first room done.

Most of your cuts will be pretty straight forward - set a chop saw at a 45 degree angle and use it to cut away the bits you don't need.  Save your remnants as they'll frequently be useful at the other end of a strip to fill in gaps.  

Since it's click lock - suggest you take your sq footage and add 30% to your order for all those funky angles you're gonna be dealing with.

On your trickier cuts, like around doors, take your time, make sure you leave the manufacturer recommended breathing room, and be prepared for this to take longer than you think because you're choosing to do it a harder way than most.  Wear knee pads too. :)

Don't forget underlayment - but you needn't run it in the same direction as the flooring.  

Looking forward to hearing how it works out!