Although you permanently can't nail down the Panel setup you can save the arrangement and, with a few clicks, reset to your layout any time things become disarrayed.
First set up your personal Panel arrangement. Then, go to Window > Workspace > New Workspace. Give it a name (let's call it Mystuff) and save it.
The next time your panels are dragged out of proper placement, just go to Window > Workspace > and choose Reset Mystuff. Everything will pop back into position.
Seconded Reset is your friend.
Or just move that character button where it's harder to drag.
You can reset workspace with an action operated with hot keys. I've never thought to try it before, but just had a go and it works fine. I used Ctrl Shift F5, but I did not check for conflicts
It's a bit like playing the piano, but it does work.
Thanks for all the feedback! That's too bad you can't lock them in place. The workspace solution is definitely a good compromise. Thanks again to everyone who replied.