I picked up a used Montana in .260 Rem a couple months ago, the previous owner had painted the stock grey with some splatter paint in black, silver and white. He did an excellent job but it's not my choice of colour. I'll either repaint it or have it dipped. What's a safe way of stripping the paint without damaging the stock? Fiberglass safe stripper?

Paint thinner or Lacquer thinner and steel wool. I stripped a paint job off of a McMillian that someone had done. I have never tried a paint stripper, but the lower power stuff like the citrus stuff might work.

If the paint is solid, I'd just clean it and rattle can right over it.

According to McMillan, sandpaper and elbow grease. That's what I did with a stock they painted (Butt Ugly Brown) that I didn't like and wanted them to repaint.

I wouldn't use chemical stripper on the stock personally...

You can use citristrip. Just don't let it sit there for long. It works well, and doesn't have an offensive odor. When the paint is all off, wipe the stock down with denatured alcohol to remove any residue

I figured that sanding would be the safest but I don't trust myself not to round off the edges and the finer areas like the barrel channel, bolt cut out, etc.

That's what I'd do and prime it first then re-paint.Chemical strippers on fiberglass might not be the best solution.

Your easiest route would be to post up in the Classifieds here and simply trade your stock with someone who likes the paint job. Win-win for both parties.