Great write up and a handy little inspiration to the tyke... and some decoration to boot.

Broke out in a sweat just thinking how long it took you to rotate those pieces in SketchUp. 

How would you cut this star, and what type of saw would you recommend for cutting small parts like this? 

My immediate though was a scroll saw (if you have one), but then I thought about my inverted jigsaw (everyone must have one of those) and not having enough opportunities, why not.  Took you model and printed the two starts I found and using a Bostik Blu Stik glued the paper to a 19mm piece of pine.  I probably should have given time for the glue to dry as it kept lifting which didn't help keeping to the line.  NO science was followed like keeping to the outside, just cut... I hate sanding so I tried to kiss the line...
without practice and the pilot run, it wasn't too shabby.

Then I thought back to my first thought,

Now let me fess up that I'm not a scroller and didn't take long to break a blade,

Had more trouble keeping a straight line than with the jigsaw, but I finished it,

I have been surprised at the finish off the jigsaw, not quite as smooth though I'm only looking at a few % out of 100.

I unintentionally probably took more care with the scroll saw as it was about 1/2 the speed of the jigsaw which gave me time to think.  
With a bit of practice, next time I would definitely try the jigsaw first. 

If your first cut is too short... Take the second cut from the longer end... LBD