Those plated diamond tools will work great for the first minute or so, less well in 5 minutes, and not at all after ten minutes of routing (if it doesn't break first). The trouble is that all the diamonds are on the outside, and once they've fallen off, there's no cutting action.
Soft limestone can be cut effectively with tungsten carbide tooling, which will keep cutting a lot longer before it gets dull. Keeping the cut wet and flushing away the stone slurry will help keep things from overheating and hurting the stone. The limestone slurry will get all over the place, and it's not good for the moving parts of your router, so protect them with plastic as best you can. Your circle probably won't be perfectly round, but otherwise the machine will most likely work okay.
Without knowing the size of the tool you plan to use, or the depth of cut or RPM range of the spindle, it's hard to give you specific feeds and speeds, but stone, unlike wood, isn't harmed by going too slowly, so get a few tools and start slow, building up to higher feedrates as you gain confidence. If the tool breaks from trying to go too fast, dial back the speed.