Aww... Generally large NN are hatchery stock, those usually are mild in temperament. does all the typical roo stuff but not aggressive to people and tolerate other roosters okay. Sounds like he has not given people any attitude? That's good news if he has not.
Assuming he is the typical mild NN roo, most likely all roosters will want to challenge each other but this would be the same as introducing a new hen in flock- there will be fighting but it eventually settles down with time. The only difference with mild roosters is the fights may be a little harder and the loser roo might be chased around a little harder than between hens but this also calms down with time. Usually there is no serious harm from fights even with LF and bantam, other than some bleeding and scabs on the face if the fight went on for more than a few seconds.
it might turn out the cochin and probably the bantam/game making the most problems- deliberately going after the big roo. Some of them are very "cocky" and think they can kick an elephant around. Sometimes aggressive behavior is given more leeway with bantams because "they are so little, ankle busters, haha" so some bantam lines are obnoxiously aggressive. Perhaps the situation will be one or both of the bantams chasing the big roo around without a fight. That happens sometimes.
I really doubt the LF NN would kill the bantam roos. Killing is usually done by roosters with game blood or from a line with non-typical and high genetic aggression. Example are some RIR lines bred strictly for egg production and roo personality is not taken in consideration at all so if there's a genetic cause for aggressive roos, it can get concentrated in the line and turn out roos that are very aggressive no matter how they are raised. This is why RIR kind of have a reputation for having very mean roosters.
LF roos can mingle with LF and bantam hens. It's surprising how bantams can handle being bred by LF and some even prefer LF over bantam roos. It will help there are LF hens so the bantam won't be getting all the attention anyways.
In short, this can work but probably have to let them fight it out and the dynamics will settle with time..
