Because of the Germa who kidnapped Sanji, the worst problem where the Germa and they start the whole mess up, ending with the alliance saving these ass from the Bmps. Without the Germa Bmp wouldn´t even think about Sanji. At the most Luffy steal a whole island from Big mom(fishmen island). Pedro sneak into BM territory years ago before the whole Sanji messup, they let him be alive duo Pekoms and he still try to sneak again. Do you expect for the Bmps to stand still and don´t do nothing? All fractions have their reason why it became a mess up.
It was a joint plot for the wedding,
It was BMs allies who did the kidnapping
They invaded Zou to do so
The BMs were the ones who were going to kill Sanji.
The BM pirates absolutely started this shit, it doesn't matter if Judge pointed them to Sanji, they were the ones who actioned this act of war.
BM chose to keep Pedro alive after taking his eye, his partner and the vast majority of his lifespan, moreso as a troll/punishment than an act of kindness.