This happens with so many DIY fixes, like lilya76 said about scraping. We try to patch things quick instead of doing them right the first time. How many other spots in the house are we just one bad seal away from a mold farm?
Is the mold really behind the caulk or could it be on the surface? I mean, interior caulk can sometimes peel from moisture but not always ruin the grout underneath. Maybe try peeling a small section first to see, it might come off cleaner than you think.
In my old apartment's shower, I had black mold spots along the caulk line. I carefully peeled back a small section like you said, and it turned out the mold was mostly on the surface. I cleaned it with a bleach solution and re-caulked, which held up for another year. In my experience, starting small can save you a lot of hassle, but your mileage may vary.