Poinciana Pruning

Have a nice Poinciana tree that gives beautiful shade over my back lawn vege garden in summer and is very popular with the wildlife

However is getting too large ( 5M high x 6M wide) and needs trimming. Have frequently cut the outer ends to consolidate but more is needed.

Will it recover from a hard prune back to the 3 main branches ? Recommendations?

Ultimately have no problem removing it but will take time for replacement to grow

Is of the large round variety rather than the low saggy branch version