Some people want to constantly move up, and others are quite happy where they are, doing their current job.  Both types are valuable but have different career development needs.

You could try explaining to your supervisor that you enjoy the work you do and feel you can contribute more in that capacity than in another role (with customer interaction).

You are probably limiting your promotion potential, but if you want to stay where you are, there was obviously a need that you were hired to fill, presumably that need remains.  Are there other (non talking) tasks you could offer to take on to enhance your current role?

At the end of the day, if it is clear that the only job available to you will be one you are not suited for and which will lead to stress and unhappiness - then I think you've answered your own question.  Tactically speaking, it is always better to seek a new job while still employed.

Good luck!

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