The Brotherhood would have almost zero problem with the Legion if they leave the Brotherhood alone. Their whole thing is gate keeping and hoarding pre-war tech and the Legion explictially bans it use.

There are cavaets to this though. They (Legion) attempt to obtain advanced energy weapons from the weapon dealing Van Graths, but are foiled by most players. And Caesar despite banning tech for his followers, indulges in it for himself. His reasons for banning it’s use by his soldiers is contextual and mostly a means of social control and discipline rather than arbitrarily ideological (in his mind). And will most likely be abandoned when he deems it nessecary, which would then obviously bring his nation to the attention of the Brotherhood. Also the Legion is hostile to the Brotherhood, you need to destroy their last bunker in the Mojave to advanced the Legion main quest.