When I look at this I see 8 pieces that it can be broken down into. I'm assuming :) that you want a large center medallion. I would take it to say Staple and have it blown up the size you want first. These should be large pieces that should be easy to either machine or hand applique.
Now this is how I would do it, others may do it differently. I would use a water soluable fusing (I don't like stiffness) or a very thin fusible around the edges of the piece to be appliqued to put on the pieces of the dogs head on the blue circle. Since its for a college I probably would machine applique with a smaller to med size zig-zag or satin stitch to secure.
Sounds like a fun project!