You could carry the yarn and do it stranded style. I haven’t looked at a strawberry lately, but it might be a good idea to do so. I know usually they don’t have many seeds on the hats, but it would be easier/better for stranding if you could put one every 6th stitch. Seed, knit 5, another seed, then skip up a ways and offset the next ones. Really… look at a real strawberry and see if it helps you with placement. There are ways to catch in the stitches to carry them farther than 5 stitches and I’ve done it. It works, but they can tend to show through a little especially if the yarns are very contrasting, red and black might work though.
If you can’t do them quite often like I said above, I think I would do them duplicate stitch. You can carry the yarn across by using your sewing needle and dividing the red yarn every little way and catching it in. Keep stretching the hat as you sew, so the seed yarn stays loose inside the hat so you don’t make the hat get smaller. I think that would show less than catching them in if you carry them a long ways stranded. But you could do a swatch trying the different methods and see what seems to look best. Swatches are always a good idea.