Without getting too far into details, these ideas get overthought and skewed depending on the media source or paper that you're reading. Consider yourself as an observer in an interial reference frame $A$, which moves at some speed $v\neq0$ relative to a different inertial reference frame $B$. Now make the further consideration that $v$ is not negligible compared to the speed of light $c$: $A$ and $B$ are in relative motion at relativistic speeds. Now consider an observer in $B$ shooting charged electrons through an electric field. We will assume that both you and the observer in $B$ have a way of detecting the motion of these electrons. You will both see the motion of the electrons as different, and you certainly could think of the electrons as having a smaller or larger charge. However, the same effect comes from a disagreement in the electric field. If we assume that the charge on an electron is a constant, then the difference in force between the two frames can be attributed to a disagreement in the strength of the electric field. This is beyond my reach, but you'd need to devise an experiment where you can measure the charge of an electron without the use of electric fields or magnetic fields. If you cannot, then it is always the case that we can attribute the disagreement between observers in $A$ and $B$ to disagreements in the electric and magnetic field strengths.
In the book Spacetime Physics by Edwin F. Taylor and John Archibald Wheeler, they state that the charge on an electron is a constant, because if it was not then you could measure the "absolute speed" of your reference frame just by measuring the charge on an electron: as your "absolute speed" increases, the charge you measure would change. This violates laws of relativity: there is no "absolute speed", just a relative one. In the same book the authors make the same argument for the mass of an object, and thus we do not think of the mass as changing. Relativistic mass "exists", but it is defined by an observer not in the reference frame of an object. In that sense it is worthless, because the observer will not handle the object ever, and the value of the relativistic mass means very little to anyone that is not in the same reference frame as our inertial observer.
Personally I much prefer to think of mass and charge not changing (which to my understanding is the prefered view), I think it makes things much less ambiguous. Some sources are not very credible, and like to elaborate improperly on ideas in relativity because it is fun and interesting. However, often these elaborations go too far and statements are made that conflict with the true principles. I think this is one of those cases where people really want to attribute a mass increase because it's cool to think of mass changing just because speed does, while in reality we only really care about the proper mass.
