The Brotherhood wouldn't stand a chance against the Legion. True, the powered armour is a strong asset, but its availability is limited, paladins are very few and far in between and they've already lost a war with the NCR. Even if every paladin manages to take fifty legionaries with him, there are plenty more waiting to enter the fight and reserves in Arizona.
That's assuming that they'd go for an honourable battle up front. Most likely, they'd simply send a squad of the frumentarii to destroy their water and food supply, destroy the entrances, create traps for power armoured troops, all echoing the Chinese counters to American T-51s over two centuries ago.
The Legion isn't mindless nor is it primitive in its tactics. Yes, they are extremely Darwinistic in their shunning of medicine and brutal in their treatment of legionaries, but they aren't stupid. Caesar's preparing well for his second battle and that includes securing artillery support, a diversion in New Vegas, securing a safe Colorado crossing by eliminating Searchlight and eroding morale in Nelson and Nipton. They aren't going to attack a well-entrenched and confident enemy, they will demoralize and weaken him first before attacking from all directions. For instance, at the Dam, apart from the brunt of the attack along the rim, there are lots of forces going in through the drainage pipes and lower parts of the installation.
Last, two other points:
1. Lack of tribal identity is explained. Caesar is very thorough when it comes to eradicating former identity. It's already been proven in history that it can be done, the more ruthless the means, the easier it is.
2. I seriously doubt that the Brotherhood has the ability to create and safely store any chemical weapons for extended periods of time. Apart from their uselessness outside the most extreme of situations, such as terror attacks against major cities, the very presence of such a weapon is an extreme danger in the confined spaces of each chapter's bunker.