I would not recommend using peat. It is too unpredictable, it will likely cause wild swings in water chemistry in a tank this small.
The Walstad method usually recommends potting soil, but personally I would go with topsoil. If you can find an organic topsoil, great, if not you will have to go with a brand like Hyponex - and it won't hurt.
These usually come in 40 lb bags and are very inexpensive, because they don't have stuff like little twigs, sticks, leaves, perlite, moss, bentonite, etc. Which is great for terrestrial plants...but will just float and be messy in an aquarium.
If you only want to get a little bag because you can't use a big bag out in your yard, get an 8 lb bag of Hyponex potting soil. It is just topsoil with perlite, really. Rinse it well to get all the little floating bits (perlite looks like little Styrofoam balls, but it's actually volcanic rock) out...and presto!