I like cutting in solid hardwood but typically make it a smaller portrait - maybe a 5x7 - where good BB Ply can be as big as your saw can handle. Sounds like you don't actually have good Baltic Birch Plywood.
1/4" oak or poplar are readily available at Lowe's and Menard's. I have not had problems with them breaking, even in pretty tricky patterns, but I'm a slow cutter. Keep a little CA Gel glue (Loctite works well) on hand in case you do get a break - it will set in a few seconds so you can keep working.
Use good and sharp blades - they are cheap so start a new pattern with a new blade and change mid-way if needed.
Here's a fairly complex pattern (Einstein) in 1/4" oak, and a simpler one (my son) in 1/4" poplar as examples. Both are 5"x7"