I think you would probably need to try a few different types first, both iron-on and normal, and you may also have to use a hoop. You will find several kinds of stabilizers at the shops that sell embroidery machines, and look for those that are wash-away and light, not heavy. They may even have some scraps you could test out for hand sewing, as the adhesive may make it difficult.
That's part of my problem. I have several types of stabilizer but I don't have scraps of this fabric to use to test since these are vintage towels I bought so I'm not sure how to test them. I just have the towels with the design stamped on them. I'm going to have to think about this.