You said that:
no laws of physics prevent the accumulation of positive charge, say, so long as the negative charge is sent somewhere else. This must be true on a cosmic scale as well.
This is not accurate. Charge conservation is important, but isn't everything - there is also the local forces that extra positive charges (for example) exert on one another. The problem is that the electromagnetic force is much, much, much, much stronger than gravity - the electric force between two electrons is is $2.4∗10^{43}$ times (24 million trillion trillion trillion times!) stronger than the gravitational force between these two electrons.
Other people asked whether we could have Electron or Proton stars, like we have Neutron starts - see
Can there be Electron and/or Proton Stars? and
Could electron "Stars" exist? and the answer is no: If a star had just one extra charge for each $1*10^{18}$ particles, the star would basically blow up instead of remaining gravitationally bound.
So for your question,
What is the most electrically charged celestial body in the universe, and what is its charge, i.e. positive or negative?
When it comes to stars, I guess the number $1 * 10^{-18}$ extra charges per particle above is an upper bound for the charge of a star made of matter and held together by gravity.
For charged black holes, the calculation is different - see https://en.wikipedia.org/wiki/Extremal_black_hole.