That's kind of the problem for the Rakshasa: he doesn't really have any great moves against a Barbarian. At least he'll be taking 23-ish points of damage per hit (he's got GWM), and I won't let him rage so he won't have resistance. Since my spell DC is 18, and he's probably got +0 to his Cha saves (I deliberately avoid learning my PCs' stats in detail so I'm not sure), there's a pretty good chance I'll be able to severely damage him or kill him without even getting my hands dirty. He'll take 140ish points of damage on average before breaking free, which is better than I could do to him with my claws.
I don't want him to run away--he's become a threat to my political plans, so I want him in prison and/or "missing" (dead).
How does the Rakshasa know about GWM? And even if he knows and has the Barbarian do it, chances are the Barbarian is not going to be doing 140ish points of damage without raging and with -5 to hit.
Rak: Dominates. One save. (15% out)
Assuming that the Rakshasa does not attack itself for fear of negating the spell, the rest of the spell ends up being:
hit 1 time: 26% out, 23 damage
hit 2 times: 39% out, 46 damage
hit 3 times: 48% out, 69 damage
hit 4 times: 54% out, 92 damage
hit 5 times: 62% out, 115 damage
hit 6 times: 68% out, 138 damage
The odds of him saving before taking 140 points of damage are greater than the odds of him saving after taking 140 points of damage.
And if the Barbarian criticals himself, he takes the damage, but also gets 10 temp hit points. So, a critical with a Sword of Lifestealing drops a few hit points from future hits.
But when the Rakshasa telepathically tells the Barbarian "attack yourself, using your strongest attacks possible", how does the Rakshasa know about raging, and damage resistance, and the sword of life stealing, GWM, etc.? Dominate Person does not give the attacker telepathic clues about the capabilities of the target.
These seems like some metagame knowledge by the DM. Either the Barbarian should be going all out, or just doing normal attacks. The Rakshasa knowing the best combo for the situation should not typically be allowed. Granted, the Rakshasa could ask the Barbarian questions before having the Barbarian attack himself, but I wouldn't have him express the answers in metagame terms. For example, would the Barbarian offer up info on "damage resistance" (or in character terms, "ignore pain when raging") when the Rakshasa asks him what his best attack is?