I don't disagree that immortality will come at the opening of the book of life...or that it occurs after the Millennium.
Putting on immortality for those in Christ Jesus actually does occur when He returns. That is specifically why those of the 1st resurrection are never subject to the "second death".
In 1 Cor.15, Apostle Paul used 4 separate Greek words in 1 Corinthians 15:53. It involves 2 separate changes in our condition, not just one.
1 Cor 15:53
53 For this corruptible must put on incorruption, and this mortal must put on immortality.
KJV
1. our flesh body must put on incorruption - that's the first required change. The body of incorruption Paul taught is the "spiritual body". The body of flesh he taught is the "natural body". The wicked dead will also have this body of incorruption, because that is the type body of the resurrection. But it is still not immortal IF... still subject to the "second death".
2. our "this mortal" (soul) must put on immortality - the "spiritual body" is our outward image, what Paul called the "image of the heavenly" in 1 Cor.15:49. It is not our personality or id. Our soul is our person, and it must be made immortal through Faith on Christ Jesus. This happens for all those in Christ Jesus, and is also why they can never be subject to the "second death".
In reality per God's Word, we ALL... are going to be in the "spiritual body" on earth beginning at Christ's 2nd coming.
In 2 Cor.5 Paul showed that already we have that "spiritual body" inside our flesh. When Jesus comes, everyone's flesh is going to be cast off, revealing their spiritual body. That is how we ALL... will appear then. For those who have died, they have already cast off their flesh body with their spiritual body image being how they manifest. So even though the "resurrection of damnation" will also manifest with their heavenly image in that future time when Jesus returns, their mortal souls will still be subject to the "second death" all throughout Christ's 1,000 years reign.
Since the idea of the "first resurrection" includes the ordinal number one, that suggests there will be a 2nd resurrection. The 2nd one isn't about the "resurrection of damnation" though, because the "resurrection of damnation" happens on the day of the Jesus' 2nd coming per John 5:28-29. The 2nd inferred resurrection at the end of the 1,000 years is about another resurrection unto eternal life through Christ like the first one. When the unsaved are judged at the end of the 1,000 years, whoever's name is found in the book of life will be in the 2nd resurrection of life unto Christ.