I found some free mouse patterns, but no good ones that were rat specific. I picked a mouse pattern and elongated the nose slightly.
I used this free Morris Mouse pattern. This pattern was pretty easy to follow. It was all SC in rounds.
--1 color worsted weight yarn
--9 mm eyes
--size H crochet hook
--embroidery thread for whiskers
--needle to sew parts together
--stitch markers (I use paperclips)
I made all of the parts before stuffing or sewing any pieces together. The original pattern had a slightly smaller nose and no whiskers. Instead of one 5 SC Rnd to start off the face, I did three 5 SC rounds. The nose is the only count change I made to the crochet pattern to make it look more like a rat. For the whiskers, I tied black embroidery thread inside the nose, and then I pulled it through and pulled the strands apart.
I stuffed and sewed all of the parts together, and I ended up with a medium sized red rat doll.
I found some other good mouse patterns. My favorite was this pattern, but I didn't think it would look as good in the red color. I might make it at some point using colors like those in the pattern. It's in Spanish, but I put the text in Google translator and the translations turned out well.
This Francisco The Fancy Mouse pattern was another one I considered making, but it would also be harder to make with this red yarn as the main body color.