There is a few problematic premisses with your question that I will address.

First and foremost, Québec was for the rapatriation of the Constitution as an idea. It was fine as long as Québec was able to get more powers or ensure its "cultural survival". The issue then was rather how it was done. In that regard, almost everything about it was problematic hence the opposition of Québec to it. Provinces did not trust the federal government to do it in a way that would please anyone but itself. Previous discussions on the Constitution (Victoria Charter in 1971 and Fulton-Favreau in the 60s) were laborious and they did not give fruitful results. Usually other provinces had issues as well but Québec was always the most extreme one.

Québec worries were that the Constitution would cement more powers in the hands of the federal government and weaken democratic tools giving Québécois more powers over their own destiny. Moreover, any discussion past or present usually required provincial populations or governments to have a "clear majority" but that consistently meant for Québec to be alone and be outvoted by the other provinces.

The issue isn't specific to Québec; most federal powers have had issues with how much powers were given to their federal government. In the case of Québec and Canada, Trudeau's rapatriation included his idea of including a charter of rights into the Constitution, something which Québec (and many Canadians) opposed. It seems a bit strange at first but the idea is that it removes powers from provinces and the people to give more power to courts and judges. Keep in mind that judges are named rather than elected and to this day, the Charters are criticised for disempowering people.

Following the Kitchen accord (named because it was negotiated in the kitchen of the Chateau Laurier in Ottawa), 9 out of the 10 provinces ended up supporting the rapatriation in exchange of a few things.

Fun fact: In French, the kitchen accord is called "Nuit des longs couteaux" or "night of the long knives" - yes, same as the famous night where the nazis eliminated the problematic people (from their point of view) inside their party.

Trudeau gave a few things to the provinces that did not support him. Initially, Ontario and New-Brunswick supported the rapatriation. The Trudeau gouvernement offered to provinces the possibility to opt-out of some federal programs while still being allowed to keep the money (in some areas only like education) and the possibility of excluding oneself of some dispositions of the Charter. They hoped the provinces wouldn't use it. The provinces gave up on the idea that they would have a veto.

This is why Lévesque was pissed: the other provinces basically just gave up a lot of their power to the federal government. The Charter gave more powers to judges named by Ottawa, they renounced the veto (no need for unanimity for Constitutional changes - meaning Québec, the usually more dissident province lost all its leverage) and lost much of a say in federal spending. Québec had wanted more powers over its social policies and they had just lost that possibility except on culture and education.

So to answer your question more directly now: well, because the British legal document gave them more leverage than the version offered by the federation. It is not a new thing neither; it used ot be that legal challenges would end up in London rather than in Canada before the Supreme COurt was created. The Privy Council in London was a lot more province-friendly than the Supreme Court was. Québec doesn't like the Queen all that much but throughout the last 150 years, you will find many examples where the Québécois preferred the English over Canadians as they found they received better legal treatment by London than by Ottawa. Québec didn't want to preserve the relic; they just preferred it over whatever was being cooked up by Trudeau and the federal government.