This is not necessarily the defining POD, but one of the first butterflies.
Robert-François Damiens alters his strike slightly resulting in a deeper gash. The wound does not kill Louis but a resulting infection does less than a week later.
A rumor (whether it is true or false) spreads that he was backed by English spies due to the current war. This infuriates the french people.
Subsequent butterflies (or desired butterflies):
A longer seven years war (not sure by how much a few years a the minimum)
France still loses in North America and India at about the same time as in OTL.
General Wolfe Survives the battle of Quebec, but is still injured. He eventually marries an American at the war's end.
France makes gains on the European continent.
Empress Elizabeth managed to live for another 6 months or so, and Prussia loses.
France's attempted/desired sea invasion of England fails or is not possible due to losing the sea battles. but in TTL both fleets are badly damaged.
By the end of the war, the other parties have dropped out made their peace, and England and France are in a stalemate, looking for peace mainly because of wartime exhaustion and expenses.
Unrest in Quebec, yet no Quebec Act:
Slightly Earlier Revolutionary war, (heavier taxes due to war expense, reduced British naval strength due to losses, and they want to rebuild it)
Cook's voyages are butterflied away. (via his death in the war or relocation or lack of ships for it)
No French Guiana 1763 settlement attempt. (Guiana is in fact lost in the war or given up in the treaty)
France still keeps a few Caribbean possessions.
1. What suggestions do you have for causing the above things to occur given the approximate POD?
2. What other butterflies do you foresee?