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I'm in need of the backstory on how a country was able to form and break out from a rich western and well developed country. I basically had to hit a reset switch in order for some of the future plans to work out. However I have problems seeing how this could happen without a civil or external war.

I would prefer the new country to have a great start, and starting off from the ruins of a war is certainly not preferable.

The only similar question I found was Can I still form a new country?, but that one focuses more on making a new country using left-over land.

The time-schedule for this is between now and about 100 years from now.

I have considered different causes such as disagreement over NATO, EU, and other memberships, but it still seems pretty thin to be honest.

Also, looking at recent events such as Catalonia vs. Spain, it seems almost impossible for a country to give up land for a new country.

Do I really need a war, or does anyone have some other suggestions?

In 1991, the Soviet Union ceased to exist as a country and the land was divided up into 15 countries, generally along boundaries of countries that existed when the Soviet Union formed in 1917, swallowing up existing countries that had been under Russian control. But this is overly simplistic and boundaries/countries were not the same as before.

Yugoslavia also broke into multiple countries around the same time (for related reasons). There were wars over this but the divisions in some cases were not the result of war but rather the cause.

In 1993, Czechoslovakia split into the Czech Republic and Slovakia.

It is sometimes known as the Velvet Divorce, a reference to the
bloodless Velvet Revolution of 1989 that led to the end of the rule of
the Communist Party of Czechoslovakia and the restoration of a
capitalist state in the country.

Vatican City, "an independent city-state enclaved within Rome, Italy" was given peacefully to the church by Italy in 1929.

And these are just a few examples from 20th century Europe. There are more in other places and times.

Even if there were not, it's something you could have in your story. The threat of war may be enough to convince a country to give up land for another country. Or it could be economic sanctions. Or the promise of economic benefit. Religion. A desire for peace. A vote of the people to secede (as Britain is (sort of) doing from the EU. Or any of multiple other reasons.

The process of balkanization is certainly very messy, but not necessarily bloody. Regions can vote themselves out via the democratic process, similar to the Scottish referendum. A colony, region or far out island territory could be abandoned for various reasons. Territories are sometimes sold like the Louisiana Purchase.

it seems almost impossible for a country to give up land for a new country.

Not necessarily. There are also example of peaceful separation: Czechoslovakia is one of these.

Czechoslovakia was a sovereign state in Central Europe that existed from October 1918, when it declared its independence from the Austro-Hungarian Empire, until its peaceful dissolution into the Czech Republic and Slovakia on 1 January 1993.

From 1948 to 1990, Czechoslovakia was part of the Eastern Bloc with a command economy. Its economic status was formalized in membership of Comecon from 1949 and its defense status in the Warsaw Pact of May 1955. A period of political liberalization in 1968, known as the Prague Spring, was forcibly ended when the Soviet Union, assisted by several other Warsaw Pact countries, invaded. In 1989, as Marxist–Leninist governments and communism were ending all over Europe, Czechoslovaks peacefully deposed their government in the Velvet Revolution; state price controls were removed after a period of preparation. In 1993, Czechoslovakia split into the two sovereign states of the Czech Republic and Slovakia.

"Nuclear Wasteland"

There are not many ways a developed country would accept such a deal. Obviously, it has to gain something in the process or it will never agree on a split…

One scenario I can imagine is the synergy between a very strong struggle for independence and a nuclear disaster. For instance, imagine that Fukushima or Tchernobyl were located in separatism regions. The day after the nuclear disaster a massive congregation of local people ask for the independence: in return they will care of their sick themselves and will act to restore their land without any help of their former country. Thinking of saving money, this may be a deal a country might accept, if the concerned region is pretty small.