Poisoning Strike does not work with a bow.
I applied the simple poison to both my short bow and my dagger to test. I was able to hit with the dagger and it applied the poison, but not with the bow. The tool tip says it should work with both.
You can apply the poison to a bow, and it will correctly display the charges. The poisoning strike skill is enabled, and if you use it there is no "The weapon is not poisoned" message, so it is correctly regiresting as poisoned.
Yet, the effect is not applied and the poison charge is not expended.