If I just stuff it with whatever filling it's just gonna be a sausage, I need the belly to be flat, and the back convex. Is there filling\stuffing that comes in sheets maybe? Preferably the one that springs back into shape well, too.
One traditional way is to cover a piece of cardboard that's cut to size with fabric and sewing that in as the bottom. This gives you a nice flat bottom to stand / sit your plushy up.
If that is too stiff for your taste, you could use fusable interfacing to stiffen up the bottom. There are many different kinds of interfacing, but you want the stiff kind (intended for heavy fabrics) for this application.
Another possibility is to add a layer of stiff fabric on the inside of the plushy. Use anything you can find that has some stiffness to it or stiffen up some leftover fabric yourself. If the plushy doesn't need to be washed, you can mix 50% water with 50% white glue or school glue or wood glue and apply that to the fabric. If you need to wash the plushy, you need to use an acrylic-based product like ModPodge, acrylic gel or varnish. Then cut the bottom piece from the stiffened fabric and from the desired plushy fabric and sew both pieces in place as if they were only one layer. Trim the seam allowance of the stiffened fabric very short to avoid weird creases.
A potential solution would be high density polyurethane foam used in furniture making. It comes in sheets that can be easily cut to size or to a specific shape with a hot knife (Normal disclaimer regarding fumes and ventilation applies) or junior hacksaw saw blade, and it comes in a variety of firmness. It also returns to shape when deformed.
Search for "upholstery foam". It's a common item on ebay.
An advantage of upholstery foam is that it can come in hypoallergenic and fire retardant versions. Making it especially safe for plush toys.
Aaargh Zombies's answer is a good idea, and upholstery foam comes in big, thick pieces, so you could fill almost any size plushie by cutting one fill piece to size and shape. It tends to be very dense and sometimes stiff because it's designed to support body weight. If that is too stiff for you, a variation on that theme would be to use low density foam.
You can get sheets of low density foam up to several inches thick. To fill the plushie with it, you'd probably need to cut a number of layers. You can cut the stuff with a scissors or sharp knife, so you could shape it without a hot knife (make each layer the shape needed for that slice).
For small plushies, you might be able to use a big sponge of the type often used for washing cars. Shape one side with a scissors for the convex back.
For tiny plushies, you could work with cheap foam kitchen sponges in layers as described above.