So one thing you can do is place limits to make it more predictable, or at least random with some predictability. You really want players to pick up and keep these negative traits, even without the permanent modifier, because it enriches the experience. Most of these aren't even that bad, but they give some personality to characters, like battle scars. How about the following:
* Each time you major mutate, you use up more mutation points. This discourages overly-mutating.
* You can only get 1 negative trait every X (say, 10) character levels, unless that trait is removed, in which case you may get another negative trait. X should be based on however many negative skills it's reasonable to have based on the max skill limit (which ideally could be raised anyway
* Removing a trait costs more skill points each time you do it.
* Those skill points don't come back.
This way, you have a choice: accept that your character will randomly get around 10 (or however many it is) negative traits over its lifetime, or, if you really hate some of the traits, fight against them with skill points. That's a real choice though, and it has consequences.
Maybe there's one particular trait you really don't want and you want it removed. In that case, it'll cost you very little. But if you keep resisting your negative traits, that will have a growing skill point cost.
Additionally, create an optional character mode with none of this logic, allowing people to remove their negative traits for free. This is the mode for people who don't want any negative traits on their character.