James,
Personally, I like handles about 1 1/2" thick. You can always glue another piece on the handle to give you more depth, like you might for a board bow.
If you're not going for too much weight and you're short on heartwood, you can make a nice shooting bow out of mostly sapwood. It may tend to take some set. Try to keep it as wide as you can to minimize this. You can also offset this somewhat by bending in a bit of reflex in the tips with steam or dry heat, and end up with a slight R/D profile.