Osage and Red Mulberry both are good bow woods.
Red Mulberry is a cousin of osage, and not as dense. Use an osage design on Red Mulberry but make the bow 15-20% wider and you should have no worries.
Red mulberry usually has thicker rings and it can be easier for a beginner to chase rings on it. In my experience, Red Mulberry appears to grow straighter than osage.
Match the right wood to your climate, then match the right design for that wood, and the rest will follow.