First of all, love your Carpenter Star. You did a great job.
My suggestion - - I would border it with the background (light) fabric so that your star is the "star" of the quilt. In other words, let it float and don't stop it with a green border. I would add enough border on the four sides so that you have a top that is the size of your bed top. The borders may not all be the same size. The two side borders should be the same. The top border should be large enough so that the Carpenter Star is not covered by pillows/pillow shams and the bottom border would then be enough to make the fit for the bed top. Then, you could add green borders for the drop on the three sides.
I also would not make the top large enough for pillow tucks. I would let the top of the quilt lie under the pillows and make pillow shams or, at a minimum, decorative pillows to match the pillow top.
Now, you have one suggestion and I am sure many more will be received. Then you have the task of wading through all the suggestions and picking the one that you like best. Good luck. Whatever you do, you will have a wonderful quilt. Please post your final quilt for us to enjoy.
Edit - looks like my suggestion is similar to "mytwopals". I was just so wordy that hers got posted while I was still writing.