Yes, to be honest, it does sound a bit too formal to me. It really depends on your office culture and the manner in which you normally speak to one another, however there is a chance you might (unintentionally) come across as stuffy or self-important, using a word like "prestige" as a self-descriptor.
The second clause, "it is of paramount importance that I succeed" is fine - perfectly accurate and grammatically correct as-is. Of course, a more natural and less formal conversational expression of this might sound something like:
"It's just extremely important to me to come out on top here."
You could even say,
"I can't emphasize enough how important this is to me."
I think the latter is likely to elicit more sympathy than contempt.
The first clause, "this is a prestigious matter," is what doesn't sound right to me. Firstly, it seems to imply that the matter itself is prestigious, when I think you mean to say that the matter is one concerning prestige - specifically your prestige, and not the matter's prestige.
You could reword this:
- This is a matter of prestige.
- This matter threatens my prestige.
But it would sound much more natural to me to say:
My reputation is on the line.
My reputation is at stake.
I feel like my character is being called into question.
I am worried my image could suffer. I don't want to lose the respect of my employees or confidence of my clients.
I'm concerned about how this will affect my professional image.
Losing could ruin my good name.
I can understand why a manager or anyone in a position of power might want to avoid revealing too much of what can be perceived as weakness by some. But when in doubt, honesty is sometimes the best policy for effective communication.
If speaking to a trusted colleague or perhaps family, put all together, I might say something like:
"This issue has been weighing on my mind heavily because I feel like
my reputation is on the line and my very character is being called
into question. Winning this means so much to me, I can't even begin to
tell you. It's just critically important that I succeed. That's why
I've been working so hard."