You already can't cast spells whilst raging. The could just extend that to "can't cast spells or use special class abilities like sneak attack". Problem solved.
Eh, not really. It's too vague, and you start having to enumerate stuff. Does it not work with Extra Attack (eg: Fighter 5/Barbarian 1)? Action Surge? Cunning Action?
If you just say, you can't add multiple damage bonuses to an attack (eg: Rage bonus, Sneak Attack bonus, etc), it works for Rage, but brings up the question of something like Sneak Attack + Hunter's Mark. Your rule starts having effects outside the specific class.
If you try to qualify it as only applying to "always on" abilities, does it apply to Rage? Because that's limited use.
If you specify
exactly these two abilities (Rage and Sneak Attack), then do you have to worry about how that rule might interact with other classes and subclasses? The more specific the rule, the less desirable. More specific rules should only be used to open up specific exceptions, if the rules are well written. They shouldn't add extra restrictions.
Maybe it should be approached from the Sneak Attack side instead? Because Sneak Attack seems to be the ability that's really causing problems for lots of other areas, rather than Rage.
The more you try to pin down, the more troublesome it is as a rule. And that's not even get into related effects, such as Dex becoming a bit of a "god" stat, and wanting to try to push away from that.
I'm sure there's a better way to handle this, but it's not something that comes easily to mind.