Even if Urgot's ultimate were altered to go through Invulnerability, it shouldn't go through Undying

The way Urgot's new ult function is that it does true damage equal to 25% of your health and then causes you to die regardless of how much health you have left. Theoretically the 25% true damage should be stopped by invulnerability but the instant death portion should go through invulnerability because it isn't based on dealing damage. Currently the way it actually works is the death effect is stopped by invulnerability as a special case.

If Urgot had a gunblade and used his ultimate on a kayle ulted target, that target would take 0 damage and Urgot would heal for 0 health because kayle's ultimate grants invulnerability, and invulnerability reduces true, physical and magical damage to 0. The target didn't die because it didn't take any damage. Then the death effect occurs, and because of the way Urgot's ult is coded it checks for invulnerability and fails to kill the target if it is invulnerable.

If on the other hand, an urgot ulted a tryndamere he would take 25% of his maximum health in damage regardless of how much health he had left and Urgot would heal for 3.75% of Tryndamere's maximum health because of the 15% omnivamp from gunblade. Its not that Tryndamere is immune to damage, he is simply immune to death itself. He is kept at 70 health because that reads easier than "gains 70 health that cannot be mitigated by grievous wounds after his ultimate expires". When the death effect triggers, the code would check to see whether Tryndamere is invulnerable and seeing that he isn't an order to die is issued to Tryndamere's character. But Tryndamere refuses the order because he cannot die.

Invulnerability and Immunity to death are not the same thing. If Urgot's ultimate were changed so that it bypasses invulnerability it still shouldn't bypass Undying Rage. Urgot's ultimate tries to kill you-- to make you experience the effect of death; Tryndamere is simply immune to that effect during undying rage, in the same sense that olaf is immune to cc during his ragnarok.

The only effect in the game that can penetrate tryndamere's death immunity (thus far) is the "Pure damage" of the fountain laser-- but that is only to prevent griefing the enemy team on their fountain when you ought to be ending the game.

Allowing Tryndamere to die from an enemy effect while he is immune to death would be tantamount to allowing Olaf to be cc'd by an enemy effect while he is immune to cc.