Why did God relent on killing Moses when Zipporah circumcised their son?

What the question is trying to do is measure two different sins with the expectation that they should result in equal judgments. When we try to do this we are making a implicit and incorrect assumption that the sins have equal degrees of profanity and require equal reactions from God's law and justice over the people under the Old Covenant.

The two situations are both grave but not equal. The first has to do with Abraham and the only external rite that was established to represent God's covenant with him. It seems that Abraham neglected for whatever reasons his duty yet in his heart may have not been as negligent as might appear, for when God was about to kill him, possibly under some disease, his wife (possibly in submission to Abraham's wish who was too weak to argue about it) cut the skin. Interpreters have various views about this because there are a few difficult passages involved but what seems to stand out is under the chastis*m*nt the heart of Abraham and his wife are corrected and the situation restored.

The second case has to do with a high profanity in disrespecting the holy instruments of the tabernacle under the Old Testament. The tabernacle was the visible pledge in those days of God's protection and kingship over his called out people. Having any disrespectful attitude towards those holy instruments was outlawed under the penalty of death. In fact the death penalty under Moses was so further established and made known for many offenses, that one answer to your question is already provided herewith. For Abraham was not threatened with death under so many laws as was developed under Moses. As Abraham was before these laws his conduct would not necessarily be judged by the letter of them.

With regard to this 'profane' high handed disrespect of the arc of the covenant it is worth looking at some language of the law regarding the death penalty. We can see the seriousness when looking at the instructions concerning the Kohathites who were basically the tabernacle manual moves, when camp was moved.

And when Aaron and his sons have finished covering the sanctuary and all the furnishings of the sanctuary, as the camp sets out, after that the sons of Kohath shall come to carry these, but they must not touch the holy things, lest they die. .....Let not the tribe of the clans of the Kohathites be destroyed from among the Levites, but deal thus with them, that they may live and not die when they come near to the most holy things. (ESV, Numbers 4:15-19)

Now with the death penalty clearly prescribed prior to this incident of gazing upon the ark with some un described profane curiosity, and because we can also understand God could have also seen deeper into a hideous degree of that profanity in the hearts of those who were gulty. And because this sin could have quickly become a disease spreading into the whole camp, being a public sin, we find many reasons why the sins may have been very different and requiring a very different correction from an infinitely wise and all knowing judge.

The bottom line is God weighs different sins, circumstances, peoples sate of heart, consequences on the public and many other things into a balance beyond our comprehension. Even from a human standpoint we can perceive differing aspects of these sins requiring some difference in an all-wise judgement of them.

The judgments of the LORD are always done for the good of the people. They are God's desperate cry for them to come back to Him so that they will not suffer by their own hands. God will not protect those who willingly put themselves in harm's way because they put themselves out of God's reach by living in sin. All that God does is for people to hear His voice and come back to Him. He is more harsh with the leaders, because they could lead people astray. this was the case of Moses.

Moses was to be the leader of the nation of Israel. People were to look at him and he'd be their intermediary. He was a type of Jesus. He was to be obedient in ALL things including the ritual for the covenant with God, which was circumcision. Moses had to see the extreme importance of this, for if he had not been circumcised, he would have been a stumbling block for all of Israel.

On the way from Midian, Moses received a startling and terrible warning of the Lord's displeasure. An angel appeared to him in a threatening manner, as if he would immediately destroy him. No explanation was given; but Moses remembered that he had disregarded one of God's requirements; yielding to the persuasion of his wife, he had neglected to perform the rite of circumcision upon their youngest son. He had failed to comply with the condition by which his child could be entitled to the blessings of God's covenant with Israel; and such a neglect on the part of their chosen leader could not but lessen the force of the divine precepts upon the people. Zipporah, fearing that her husband would be slain, performed the rite herself, and the angel then permitted Moses to pursue his journey. In his mission to Pharaoh, Moses was to be placed in a position of great peril; his life could be preserved only through the protection of holy angels. But while living in neglect of a known duty, he would not be secure; for he could not be shielded by the angels of God. In the time of trouble just before the coming of Christ, the righteous will be preserved through the ministration of heavenly angels; but there will be no security for the transgressor of God's law. Angels cannot then protect those who are disregarding one of the divine precepts.

Patriarchs and Prophets, p. 255-6

As far as the people of Bethshemesh, this paragraph explains it best.

The men of Beth-shemesh quickly spread the tidings that the ark was in their possession, and the people from the surrounding country flocked to welcome its return. The ark had been placed upon the stone that first served for an altar, and before it additional sacrifices were offered unto the Lord. Had the worshipers repented of their sins, God's blessing would have attended them. But they were not faithfully obeying His law; and while they rejoiced at the return of the ark as a harbinger of good, they had no true sense of its sacredness. Instead of preparing a suitable place for its reception, they permitted it to remain in the harvest field. As they continued to gaze upon the sacred chest and to talk of the wonderful manner in which it had been restored, they began to conjecture wherein lay its peculiar power. At last, overcome by curiosity, they removed the coverings and ventured to open it.

All Israel had been taught to regard the ark with awe and reverence. When required to remove it from place to place the Levites were not so much as to look upon it. Only once a year was the high priest permitted to behold the ark of God. Even the heathen Philistines had not dared to remove its coverings. Angels of heaven, unseen, ever attended it in all its journeyings. The irreverent daring of the people at Beth-shemesh was speedily punished. Many were smitten with sudden death.

Patriarchs and Prophets, p. 589

As for Uzzah, he was punished for his distrust in God and his impatience.

The ark remained in the house of Abinadab until David was made king. He gathered together all the chosen men of Israel, thirty thousand, and went to bring up the ark of God. They sat the ark upon a new cart, and brought it out of the house of Abinadab. Uzzah and Ahio, sons of Abinadab, drave the cart. David and all the house of Israel played before the Lord on all manner of musical instruments. “And when they came to Nachon’s threshing-floor, Uzzah put forth his hand to the ark of God, and took hold of it, for the oxen shook it. And the anger of the Lord was kindled against Uzzah, and God smote him there for his error; and there he died by the ark of God.” Uzzah was angry with the oxen, because they stumbled. He showed a manifest distrust of God, as though He who had brought the ark from the land of the Philistines, could not take care of it. Angels who attended the ark struck down Uzzah for presuming impatiently to put his hand upon the ark of God.

The Truth About Angels p. 126