As titled, can holding a constant static charge affect the CNS? If so, is it detrimental? What are the short/long term effects, if any?
In the absence of research, I will give a few physics observations.
Generally, voltage is considered to be relative. So electrical engineers for example completely disregard static charge in most engineering applications.
The exception where static charge is relevant is when there is a risk of discharge, i.e. a discharge shock. That can damage equipment. It's known as well that electrical currents can damage tissue and cause cardiac arrest. So for example if you are electrically charged, then discharging in a way that passes significant current through the heart could be dangerous. However static charges would be small in day to day tasks, because of the small capacity of the body to hold the charges and high internal resistance (a greater capacitance would mean more charge at the same voltage). Maybe holding a highly capacitive object (such as a literal capacitor terminal) could change this consideration.
But the static undischarged charge itself is unlikely to have any physiological effects at all because of insensitivity of most materials and processes to absolute potential (some exceptions include semiconductors). The electric field inside the body should be negligible and charge would concentrate on the skin. The physics of internal processes stays largely the same whether you are at absolute +1000V relative to Earth (ground) or at +0V relative to Earth, although in the first case the discharge could lead to a nasty shock (likely not lethal if you're floating/not connected to any power source, probably lethal otherwise).