Okay. You say so. Please provide the quotes from prophecies that speak of the "period of peace" that you are referring to. I want to read the description of that period in those prophecies that you are talking about.
If such details existed, we wouldn't be having this debate.

The prophecies outline a framework of what will happen. Common sense (and Church history's prior examples) fill in the gaps.
You say "the peaceful time of Constantine." Do you know what you are talking about?
Heard of the Arian Crisis?
https://en.wikipedia.org/wiki/Arian_controversy
How about the Donatists?
https://en.wikipedia.org/wiki/Donatism
All of these problems occurred during this "peaceful time" in and around Constantine. So if that is your definition of the "period of peace," then I don't disagree with you that it can happen in the end times. We are already there. But that is not the description of the "period of peace" in Holzhauser and other apocalyptic prophecies.
Compared to 300 years of being hunted down by Roman officials, treated like 4th world citizens, being fed to lions, and killed in various torturous ways, yes...the times of Constantine were quite peaceful.
Augustine explains that as long as the Church is on the "earthly pilgrimage" it will be a mixture, containing both the elements of the Earthly City and the City of God within it. Only upon Christ's return will the church ever have true peace. This is lack of peace is why were call the Church on Earth the "Church Militant."
I agree. I don't know where you heard that this "certain period of peace" which Our Lady of Fatima mentions is supposed to be some utopian dreamland where every person is a walking saint, where it never gets cold and puppies don't die.
The prophecies say the following, none of which is outlandish, or beyond reality, but is quite consistent with a restoration of the Church hierarchy and normal, human ways of restoring societal and ecclesiastical order.
1. When Russia is consecrated by a true pope, then Russia will convert and start defending the Church.
a. Many prophecies say this will happen during the midst of a war, which means that Russia will then start attacking the bad guys.
b. The war will be won by Christians and, due to the miraculous nature of this victory (similar to the miraculous victory of Constantine), the whole world will know that Our Lady is the cause, as God wills to honor Her and to have non-catholics recognize that She is the reason for the end of the war.
2. This dramatic end of the war and the dramatic conversion of Russia will cause countless non-catholics to convert to the Faith. (Much like the miracle at Guadalupe converted 15 million souls).
3. The end of the war, which was fought by the Great Monarch-in-waiting, will coincide with the global disillusionment with the democratic form of govt (which is, on the whole, only a 200 year experiment) and a return to a Monarchical govt, worldwide.
a. The democratic/republican/rule-of-the-people govt will be seen as the vehicle for the war, as it enabled the elites to corrupt the system and take control.
b. No form of govt is perfect, but as history shows, when a govt fails, the people often demand the opposite. So, naturally, and also consistent with Church history, a return to a monarchy will coincide with a return to Church rule.
4. What follows during this "period of peace" is nothing more than a total, societal revamp/rebuild.
a. A rebuilding of Church order, structure, clergy, schools, churches, etc.
b. A Church council is called to denounce the errors of V2, protestantism, socialism, materialism, etc
c. The Church grows due to the influx of converts. There is a renaissance of energy, piety and devotion.
d. Religious orders are restored, new ones begin and the religious life is renewed.
e. Family life is restored, because society is at peace and ordered, through the Church.
5. The global monarchy, as ruled by a Holy Roman Emperor, who works with the Angelic Pope, works to restore order throughout the world.
a. Nations which had never been catholic before (i.e. Middle East, Asia) are sent missionaries and churches/schools are built.
b. The whole world, generally speaking, is catholic. (Not every single person).
c. There is order, there is peace, there is the worship of the Faith in all countries.
This lasts for anywhere between 50-100 years. Much like the "peace" of the 13th century. There is nothing in the above that is fantastic, other than Our Lady's victory, which will be miraculous. But the rebuilding of Catholicism and the work that it will take to restore the Church, will be normal, human efforts. All of which are consistent and predictable according to history, which proves that after a political system (i.e. democracy) fails in utter disaster, that the opposite form of govt (i.e. monarchy) typically takes its place. This is just human nature.