I realised that the ground fabric was too flimsy, and started again on heavier fabric. This was much more successful.
The outer white edge was done in white crochet cotton in stem stitch. The blue was done in cotton perle thread in stem stitch. I used stem stitch because I wanted this area to stand out and be clearly defined.
I wanted the badge to stand out and not be too flat, so I added an extra, smaller piece of felt on the back. Then I edged the whole shield-shape with some silver gilt thread.
Here is the badge sewn on the recipient's new garb. The recipient is the perfect example of all the courtly graces, so I was very pleased that the badge looked so nice and was received so well.