"Avoid or overcome" ........Hmmmm.........let me see.........
......don't fight them? Walk away? Don't try to punch, kick, push, or throw them down!
If it is genuine Aikido they are using, they are not going to attack you (except with a feign or distracting blow). The main goal of Aikido is to avoid the fight, and avoid being overcome, so what you are asking is "how do I avoid someone who is avoiding me?"
If you attack a skilled Aikdoist, or Hapkidoist, the main defense will be to avoid your attack, redirect you force, and use your energy, and lack of balance against you. To avoid this, don't attack. Don't extend energy forcefully. If they attack you, they are not following Aikido principles, and you can skillfully (and cautiously) use their force against them, or counter-strike quickly without sacrificing your equilibrium.
Just out of curiositiy, why would you ask about "Tae Kwon Do vs. Aikido." Is this one of those, "which one would win in a fight questions?" If so, don't ask. You know the skill of the practitioners involved, and their intent, or aggression is going to be the main factors. As a Taekwondoist, if I want to defend against an Aikidoist, I walk away. If they attack, they have sacrificed their advantage, and I can counter with a variety of short low kicks, and rapid hand strikes that do not commit to any forward aggression or powerful circular movements that they can utilize. Then, back away and wait for their next mistake.
CM D. J. Eisenhart