Hi,
I think it would be easier and neater looking if you do as you say and cut the letters out of the background fabric then sew that onto white fabric. This would give you the same consistent pattern on each mat. To cut the letters out separately would be nice and you could outline them in a decorative stitch, but this would take longer. However, you can just as well outline each letter on the background fabric that is sewn to the white and get that same look. Good luck with your project!