Water-damaged briar finish - advice needed

[Nature]

Similar to a water ring on furniture, I would try a buffing first. If that is not enough, then start with a very fine sandpaper, like a #1000-1500 in the affected areas followed with buffing. I have used the back side of the sandpaper to polish wood before final buffing.

EDIT:
As an afterthought addendum, it appears as there is not a clear lacquer layer? It looks like an oil rubbed or stained finish. If just light superfine sanding/ buffing is not enough, you may need to finish with an oil rub (Tung or similar) and possibly a matching stain. Linseed oil takes on that orangeish hue over time. Both Tung and Linseed will have an odor initially, but dissipates with time as it cures. Neither should absorb beyond the surface layer to affect smoking. Other options could be waxes commonly used to finish pipes and wood, like carnauba.

Perhaps someone with more experience of finishing pipes specifically could chime in?