Lot was saved the whole time.....you do not understand the verse.....vexed means he ,that is Lot was worn down,by the filthy conversation of the wicked
He practiced some of their same wickedness, We see how much he was in this in
Genesis 19:
6 And Lot went out at the door unto them, and shut the door after him,
7 And said, I pray you, brethren, do not so wickedly.
8 Behold now, I have two daughters which have not known man; let me, I pray you, bring them out unto you, and do ye to them as is good in your eyes: only unto these men do nothing; for therefore came they under the shadow of my roof.
He was so into this he offered his daughters to these men for their pleasure.
We see just how much Lot was involved in verse 14 And Lot went out, and spake unto his sons in law, which married his daughters, and said, Up, get you out of this place; for the Lord will destroy this city. But he seemed as one that mocked unto his sons in law.
He was seen as a hypocrite by his son's in law.
Then he allowed his daughters to get him drunk and we see,
30 And Lot went up out of Zoar, and dwelt in the mountain, and his two daughters with him; for he feared to dwell in Zoar: and he dwelt in a cave, he and his two daughters.
31 And the firstborn said unto the younger, Our father is old, and there is not a man in the earth to come in unto us after the manner of all the earth:
32 Come, let us make our father drink wine, and we will lie with him, that we may preserve seed of our father.
33 And they made their father drink wine that night: and the firstborn went in, and lay with her father; and he perceived not when she lay down, nor when she arose.
34 And it came to pass on the morrow, that the firstborn said unto the younger, Behold, I lay yesternight with my father: let us make him drink wine this night also; and go thou in, and lie with him, that we may preserve seed of our father.
35 And they made their father drink wine that night also: and the younger arose, and lay with him; and he perceived not when she lay down, nor when she arose.
36 Thus were both the daughters of Lot with child by their father.
He had children by his daughters and one of those sons peoples were quite prominent in the area and the Moabite people came about the same Moabites who hated the Jews but that Ruth came out from.
Lot was fully involved in sin in Sodom and yet was called righteous by Peter, why was Lot righteous? Because he had the same faith as Abraham and all Believers a faith in the Savior who bought Him with the same price He has bought all believers, that is Jesus Christ.
All who are truly saved are seen in what Peter stated in 1 Peter 1:
18 Forasmuch as ye know that ye were not redeemed with corruptible things, as silver and gold, from your vain conversation received by tradition from your fathers;
19 But with the precious blood of Christ, as of a lamb without blemish and without spot:
We are redeemed by the precious blood of Christ not by sel-righteous works added to that Faith.
Hebrews 9:12 Neither by the blood of goats and calves, but by his own blood he entered in once into the holy place, having obtained eternal redemption for us.
His blood paid the price for our redemption thus salvation comes by Grace through Faith in that Savior not by anything we do to keep ourselves saved.
Revelation 1:5 And from Jesus Christ, who is the faithful witness, and the first begotten of the dead, and the prince of the kings of the earth. Unto him that loved us, and washed us from our sins in his own blood,
By His blood all of our sins are washed away and we are saved and not one of the passages above says we must keep ourselves saved but we are saved by HIS BLOOD shed for our sin and because we called upon Him for that Salvation
Romans 10;13 For whosoever shall call upon the name of the Lord shall be saved.
What will our righteous do for us, Isaiah 64: 6 But we are all as an unclean thing, and all our righteousnesses are as filthy rags; and we all do fade as a leaf; and our iniquities, like the wind, have taken us away.
How can those filthy rags help restore anyone who was once saved by God's grace through Faith, they cannot. Christ died once for all the just for the unjust and as such only He can remove salvation from those whom HE Freely gave it because of their faith otherwise man must use filthy rags of self-righteousness to keep himself saved.