I've never heard of this species before, so I did some internet research.
Tipuana tipu is the scientific name of the species and the sole species in the genus
Tipuana. It is closely related to the genus
Pterocarpus in the Fabaceae family. This latter genus is best known for the commercial wood species African Padauk (
Pterocarpus soyauxii), and also Muninga (
Pterocarpus erinaceus). Since I'm not familiar with
Tipuana tipu wood, I can only compare it to its closest cousins; species in the
Pterocarpus genus. These species tend to have relatively short fibers (something I observe in many Fabaceae species), and often have interlocked grained (visible as cross-grain in sawn boards). The presence of cross-grain in
Tipuana tipu is to be expected, based on
a report of the poor splittability of the wood. Wikipedia states the wood as "whitish in colour, strong and fibrous", which sound pretty good for bows. I can't find much more about this wood.
This site lists the specific gravity as "Density 0.67–0.75 g/cm³".
I think you should give the wood a try. It might be difficult to split a stave from a log, but you never know. Remove the sapwood and go for a heartwood ring. A backing, applied to a milled board, would perhaps be better than a selfbow.