There isn't anything about a greatsword wielding barbarian that says "Oh yeah this is an assassin"

He should be a lumbering force of nature who is very easy to outrun or outmaneuver but who is unassailable in his element. He should not be a skulking rogue that specializes in strike fast-first methods and dismantling towers. His true threat should come from the fear he imposes on you when his allies are near to prevent you from outsmarting or outmaneuvering him. In other words, big messy bloody battles are the things that barbarians dream of. They don't dream of demolishing structures and skulking around like a coward.

Here is what I suggest:

Undying rage now gains +X seconds on kill or assist, but tryndamere is self silenced during.

He can't use q w or e during undying rage. He can still use summoners and items.

He gains something and he loses something:

What he loses: Now he's not nearly as oppressive in a split push because his ultimate prevents him from escaping with spinning slash, and prevents him from using mocking shout so he can't use it to help him dive you under the tower. Kiting him in his ultimate is much easier and you don't need to burn a summoner spell or multiple allies to do it, because he can't use skills but you can.

What he gains: He is a better team fighter, but only significantly so if his team can burst down targets or kill people at a steady pace. He is pretty much immune to death for the entire team fight as long as his team keeps churning kills out on your team because his ultimate will keep gaining time. He still probably won't do much damage directly to squishies himself though because he can't use skills to slow or close the gap on a squishy target. He really can't be expected to reach an important target without a ton of help from his allies in terms of speed boosts disables and positioning. What he becomes is a tank who's tankiness in a team fight depends entirely on scoring assists and who trades 100% of his utility and mobility for raw damage output.

X would probably be something like 3 seconds.

Do you think he'd be more interactive with both the enemy team and his own team this way? Would he offer his allies the experience of slowing, snaring, and dooming their enemies for the lumbering devastation of the ponderous blows from the freljordian juggernaught? Would he have more reason to fight alongside his teammates and offer his enemies the experience of successfully peeling away from him and damning him to die in a heap of malcontent? Would he be more interactive this way?