Just for general knowledge, my pen had similar problems. I dismissed it being a battery issue as it was a new device less than a month old.
I went to the MS store and assured the rep that it could not possibly be the battery. I only let her put in a fresh AAAA battery just to "check that box" in the troubleshooting steps list so we could get on with the replacing of the pen.
Well, screw me the fresh battery worked and that was that.
I have now been using that pen/battery for almost three months and there have been no further errors or problems with it at all.
So even though it is impossible that it could be the battery...it just might be the battery.
That's my story and I'm stickin to it.