An opponent of this demeanour can be viewed as a curse or a blessing.
As you've probably figured out by now, the development of mental strength represents one of the greatest demands of martial arts, and one of its greatest benefits.
Regardless of whether you like your sparring partner or not, the opportunity to spar with someone who is better than you is often an opportunity to learn. Even if this person is less interested in teaching you than in simply dominating you and being cocky and rude, ask yourself:
How can I benefit from this situation? How can I use this situation to become a better fighter?
If a person can get 'inside your head', it's nothing to be ashamed of. It's a proven technique that many fighters - perhaps most famously Muhammad Ali - have used to overwhelm some of the toughest athletes the fight game has produced.
Think of it as any other technique; as an attack you need to become conditioned to; to become able to defend. Just as you can learn to process pain, and to respond to physical assault, so you can learn to toughen your mind; to become mentally resilient.
Next time you encounter this person, do so less with a sense of dread than with a resolve that you won't let his bravado put you off. Remind yourself that it is merely another form of attack you need to learn to offset.
It's not an easy task. I'm not suggesting this should come naturally to you and that you can transform this situation over the course of one three minute round. Don't try to. You have identified a weakness in your arsenal and this is a great thing, because now you can learn to address it, session by session, in the same way that you would hone your other defences, like parries and checks and catches.
If you have a quick wit, you might try talking back, but if this is not your forte, you will likely just be playing into his hands.
Focus on your breathing, and on keeping your emotions in check. Don't let his barbs draw you in; to dictate the flow of the fight. Try to regain the initiative. Try strategy and tactics. Think about his strengths before your next session and research ways of countering them. Speak to other good fighters. What would they do? If he's way out of your league, so be it.
Try attacking when he's in the middle of talking at you, as it is harder for someone to cognitively balance the act of speech with the act of precise and rapid physical movement.
Set yourself attainable goals. If it's primarily the mental game that's putting you off, then that is where you need to direct your efforts. Aim to complete a session more calmly than you have in the past; focus less on his words and eyes than on his torso (and peripherally his limbs). Until your mind is calm, you will likely be unable to fight at anywhere near your capacity. Concentrate on making it through the round with composure, poise, resilience and above all, determination. Congratulate yourself on any small successes you experience in this regard, because they are vital to your growth as a fighter and because you can draw upon them during your next encounter.
Also, remind yourself that your situation also has real-world relevance. The mouthy sparring partner sometimes has a lot in common with the drunken thug who likes to start fights in public. The mental gains you make in the gym can translate very well to actual conflict.
He may be a better fighter than you. So what? There's probably hundreds of thousands or even millions of fighters better than you. Most of us are in a similar situation. Give yourself time and come to see him as a resource; as a unique training opportunity that no-one else in the gym is providing you.