In early drafts of the story, Ursula is supposed to be Triton's sister, that may be the influence as to why Triton couldn't kill her (Maybe Poseidon would punish him). The opening song had deleted a line about Ursula and Triton being siblings.
What ever you do, Don't fall asleep

Because Triton is not a murderer
I really think its a simple as that, if she had threatened Ariels life or his life he probably would have fought her and killed her in self defense (like Eric did later when she was about to kill Ariel) but what she did at that point was legaleven though it was despicable.
She might have manipulated Ariel into signing the contract, but she still signed itit would have been murder if Triton had killed Ursula just to get his daughter out of a contract she willingly signed. By law Ariel had bound herself to Ursula and Triton would have had to break the law to get her out of itand how honorable a King would he be if he were to break the law to bail out his daughter?
Besides for the sake of the story Triton had to redeem himself for his previous actions and Ariel had to learn something too about facing the concequences of her own actionsit was a turning point for both those characters in the film.
Nailed it.

Yes, but we are talking about the movie

In the Broadway show ARIEL gets (kills) Ursula
Smashes her shell

Because Triton is not a murderer
I really think its a simple as that, if she had threatened Ariels life or his life he probably would have fought her and killed her in self defense (like Eric did later when she was about to kill Ariel) but what she did at that point was legaleven though it was despicable.
She might have manipulated Ariel into signing the contract, but she still signed itit would have been murder if Triton had killed Ursula just to get his daughter out of a contract she willingly signed. By law Ariel had bound herself to Ursula and Triton would have had to break the law to get her out of itand how honorable a King would he be if he were to break the law to bail out his daughter?
Besides for the sake of the story Triton had to redeem himself for his previous actions and Ariel had to learn something too about facing the concequences of her own actionsit was a turning point for both those characters in the film.

Technically, although the contract is indeed indestructible for Triton's abilities to directly penetrate, any abilities Triton uses COULD force her back. And considering the only way to actually destroy the contract is to kill her, there was a way to destroy the contract. Namely, have Triton blast her when she tries to use the contract to shield her, then bend the beam so that it would force her right onto one of those spears hanging off the wedding ship. Sort of like how Goku managed to beat Cooler here: https://youtu.be/2vdrCwDChwI?t=184 or the Son family killing Broly here: https://youtu.be/wyt_KYjQGKI?t=185
But I can forgive this bit. After all, Triton needed to redeem himself of his jerk behavior towards Ariel, and Ariel still needed to undo the mess she in part caused.

For over two decades many people have come up with reasons to criticize this movie as opposed to other Disney films BUT some times a valid criticism emerges
The Contract is Ursula's shield so that could protect her from such an attempt
Another possibility is Triton does not want to stoop to the level of a humans barbarism
It should be interesting to see reaction to this post