You may want to try a spiral downcut bit. The way the upcuts work is they pull the mateial from the bottom up and sometimes cause splinering at the top, good for mortising because it keep the hole clean. The downcut bits cut from the top down and will leave a smoother top surface.
When the upcut bit is "pulling" the wood up, it is probably breaking off your sensitive point. If you used a downcut bit with a solid flat surface, you will be "pushing" the wood down and the tip should not break so easy.
Turning the material upside down may work as well and worth trying before buying anymore tools.
Hope this helps.