4 Kings vs. Mashiah Part 1
Bereishit Rabba 42:1
ומצאו יושב ודורש...
הפסוק הזה חרב פתחו רשעים וגו' זה אמרפל וחביריו
להפיל עני ואביון זה לוט לטבוח ישרי דרך זה אברהם
חרבם תבא בלבם ויחלק עליהם לילה הוא ועבדיו ויכם
The Nezer Hakodesh ד"ה להפיל עני ואביון explains this Midrash to say that the main purpose of the 4 kings attacking the 5 cities of the plain was to attack Avraham and Lot, even if the original purpose of the war was to quell the rebellion - it quickly turned into a fight against Lot and Avraham. So they captured Lot and they knew that Avraham would go after him, so that's how they were to get both of them.
The Nezer Hakodesh asks 1) how they knew that Avraham would go after him, and 2) in fact why did Avraham go after him if Lot was such a Rasha that his shephards were thieves? Also, the Zohar says that Lot had an appearance similar to Avraham's so by taking Lot, they were satisfied and left Sedom, but 3) how is taking someone with Avraham's appearance good enough for the 4 kings if they really wanted Avraham?
The Nezer Hakodesh answers these questions by saying that since Amrafel was really Nimrod, he was very intelligent in כח הטומאה - the dark forces. He knew that Avraham's descendants would hold the key in destroying these dark forces, and that's why Nimrod wanted Avraham dead so badly. However, Nimrod also knew that the final Tikkun against the dark forces would only come via Mashiah, who came from Lot (via Mo'av -> Rut -> David). This is why Avraham went after Lot to save him, and how they knew that Avraham would go after Lot to save him. Also, if a person is similar to another in appearance, it means that their spiritual souls are similar as well. Even though Lot was no Avraham, but hidden within him was the potential for a later generation to be Mashiah, and that's why his appearance was similar to Avraham's. This is what Nimrod saw as well, and that's why he was satisfied with Lot being taken.