Grounding a POE-powered device
Most probably it is not a device charged, it is you who is charged. Look for the cause of charge around you: your clothing, carpet, material of your sofa...
If device would be charged, when you touch it, you would synchronize with it very quickly. Here you say you need to wait for several seconds to level the charge - for me it means extended charged surface with limited resistance and high capacitance - which is you...

Update: Capacitive touch screens are widely used in handheld devices, and these devices are usually not connected directly the protective ground, and user usually touches chassis of the gadget before touching the screen.

The issue you raised is not only about proper operation of your touchscreen, but also lies in the domain of human safety. If you let user touch grounded chassis of the screen, or step onto the metallic plate with bare feet, it may cause shock or at least unpleasant feelings if user will appear highly charged (e.g. due to natural reasons - wearing wool clothing).

What I would start with is to make metallic plate around the screen, and connect it to device's protective ground using circuit similar to shown here http://www.ti.com/lit/an/snoa464c/snoa464c.pdf on page 16 near "Bob Smith termination" text, or just put 2kV 1000pF capacitor between this plate and protective ground (as it is designed in integrated RJ-45 jacks without PoE) - to the ground wire of splitter as you proposed. Then, I would require users to touch this plate before they touch the screen.

In general you should take care about proper grounding of all devices in your environment (where possible), so that there would be no cases when user touches two devices with different grounding or potential, or touching grounded and ungrounded devices.