How is good news of military victory for the Romans a message that would have emanated from the Jews?
Evangelion means simply "good news." The Greek comes from aggelion which simply means "message or news." (Note that two gammas together in Greek are pronounced as "ng.") The word in question has an "eu-" prefix which simple means "good." Thus, it is "good news."
Always start with the simple meaning. If it is good news of a military victory, that will be apparent from the context. In the NT, the word appears 77 times in 74 verses. According to Strongs, it is translated as
1) a reward for good tidings
2) good tidings
a) the glad tidings of the kingdom of God soon to be set up, and subsequently also of Jesus the Messiah, the founder of this kingdom. After the death of Christ, the term comprises also the preaching of (concerning) Jesus Christ as having suffered death on the cross to procure eternal salvation for the men in the kingdom of God, but as restored to life and exalted to the right hand of God in heaven, thence to return in majesty to consummate the kingdom of God
b) the glad tidings of salvation through Christ
c) the proclamation of the grace of God manifest and pledged in Christ
d) the gospel
e) as the messianic rank of Jesus was proved by his words, his deeds, and his death, the narrative of the sayings, deeds, and death of Jesus Christ came to be called the gospel or glad tidings
Now, let's look at the Tanak verses you give for the Septuagint ("1 Samuel 31:9 ; 2 Samuel 18:31 ; 1 Kings 1:42 ; but cf. 1 Samuel 4:17").
1 Samuel 31:9 They cut off Saul’s head and stripped him of his armor. They sent messengers to announce the news in the temple of their idols and among their people throughout the surrounding land of the Philistines.
2 Samuel 18:31 Then the Cushite arrived and said, "May my lord the king now receive the good news! The Lord has vindicated you today and delivered you from the hand of all who have rebelled against you!"
1 Kings 1:42 As he was still speaking, Jonathan son of Abiathar the priest arrived. Adonijah said, "Come in, for an important man like you must be bringing good news."
In all three of these verses, words from evangellion are used to translate the Hebrew verb basar, which simply means "bring a message, tell news, or give tidings." A noun form of the Hebrew word appears in 1 Samuel 4:17 (which you list as "compare for contrasting") and the messenger brings bad tidings.
1 Samuel 4:17 The messenger replied, "Israel has fled from the Philistines! The army has suffered a great defeat! Your two sons, Hophni and Phineas, are dead! The ark of God has been captured!"
Forms of basar appear 24 times in the Tanak. It is often used of a military victory, but that is not required. For example, in Psalm 40:9, the poet "preached righteousness" (see also Psalm 96:2, Isaiah 40:9, Isaiah 60:6, Nahum 1:15, and others). A very important verse is Isaiah 61:1.
Isaiah 61:1 The spirit of the sovereign Lord is upon me, because the Lord has chosen me. He has commissioned me to encourage the poor, to help the brokenhearted, to decree the release of captives, and the freeing of prisoners,
This is important for two reasons. 1) basar is translated by euaggelion. 2) Jesus reads this verse at the commencement of his public ministry in Luke 4:18, 19.
Luke 4:18 "The Spirit of the Lord is upon me, because he has anointed me to proclaim good news to the poor. He has sent me to proclaim release to the captives and the regaining of sight to the blind, to set free those who are oppressed, 4:19 to proclaim the year of the Lord’s favor."
What makes this significant and detrimental to Atwill's argument is that if Jesus had read on from Isaiah, the context does become something of a military victory, the day of the Lord's vengeance. Jesus stops in mid verse because He is not there for a military victory at this time (that comes later).
In short, if the message is about a military victory, then euangellion is referring to a military victory. However, the same word could be used to say that I got a raise or my son got out of the hospital. It's all context.