The solution would be to detect attempts at cutting you off and automatically close the gap momentarily to discourage this behavior.
Ideally your autonomous car would also take part in the community of tailgaters and communicate forward that the current speed is unacceptable if this is the consensus of several tailgating vehicles, and gently reduce distance in a controlled way until the car in front gives in and speeds up or yields.
What you're describing is a car that eschews defensive driving in favor of offensive driving. Speeding up to keep people from cutting in front of you only works until the other driver doesn't see (or doesn't care) and now you're in a crash. Or traffic suddenly slows as you're speeding up to close the gap, and now you can't respond quickly enough and now you're in a crash. Letting someone cut you off and slowing to return to a safe following distance is by far the safer option.
There's a reason it's called a safe following distance. Anything closer and you cannot stop in time. Even temporarily. Especially when autonomous cars in front of you may be suddenly braking as well.
Safer from the minimizing personal liability perspective. If cutting people off is a valid tactic because they bend over and take it then more people will do it more of the time and that makes everyone less safe.
This is basically the opposite of the ideal solution.
The solution is to remove dangerous drivers from having control of 2-ton deathtraps, not contributing to the same behavior that is causing the problem.
Opportunistic, unsafe behavior is exactly the reason we need autonomous vehicles, rather than human-controlled vehicles.
Then you'll have AI engaging in unsafe driving habits because of their drivers and people will blame AI for every single accident rather than take responsibility.
Good way to get self-driving cars banned before they take off.
Ideally your autonomous car would also take part in the community of tailgaters and communicate forward that the current speed is unacceptable if this is the consensus of several tailgating vehicles, and gently reduce distance in a controlled way until the car in front gives in and speeds up or yields.