If you do a search here on this Board ... there's been several Carpenter Stars posted as full size bed quilts.
Also, you should find a link to a Double Carpenter Star pattern which you might find will help you deal with the white space you aren't liking.
This pattern has also been done in smaller block sizes, with each Star becoming one block, then putting several together for a quilt
Remember too, you can change your block size, which might help minimize the background for you, and increase the visibility of the colours in the star.
Or of course, Google ... and you'll find LOTS of info too!