Doing some armchair research as I can't go exploring any more. My late father told me a long time ago that there was a limekiln on the north bank of the Forth at NS807952, and this is confirmed by a number of the older OS maps. There was a pier slightly downstream. Now they would need to have brought coal in by boat, nearest available source would be any of the Clackmannanshire mines via Alloa docks, no Manor Powis colliery in the 1800s. There appears to have been a road crossing the railway where the modern level crossing is. Economics would be very much against bringing in the limestone by boat or horse and cart, and there is no indication that the railway ever had a siding or unloading platform there.
Now when I was very young I lived in the area and when visiting either of the grandparents if I was bored I would wander along the footpath at the base of the Abbey Craig, behind the wall. All kids used to go that way, much more interesting and safer than being too close to the road. I remember noticing at least one feature which seemed a bit odd, and thinking about it now, I think it was the approach cutting to a mine entrance, just like those at Murrayshall (where one became uncovered a few years back). There would be quite a few tonnes of boulders covering the entrance.
I have just checked the geology and because the whin sill which forms the bulk of the Abbey Craig "transgresses" the carboniferous strata in several places, the limestone below the sill is not the same as the Murrayshall limestone, but a limestone nonetheless, and it ought to be exposed. I actually remember seeing a softish shale at the base of the whin behind Causwayhead park, and it is known that sandstone was quarried at the north west corner of the park to build the Wallace Monument. So we know that suitable carboniferous strata are there.
http://mapapps.bgs.ac.uk/geologyofbritain/home.html confirms plenty of limestone.
The problem is that the area is no longer maintained. The council used to bring down any loose boulders annually in the interests of safety, so there was little fear of a rock fall. I do remember a huge boulder in the middle of the A907 road in about 1954. It was there for several days, till they used explosives to break it up. But around the time I left the area, 1973, the council gave up on maintenance and attempted to restrict access, so the footpath eventually became overgrown. Google Streetview shows what looks like a jungle, so it may be very difficult to get in there and have a good look for signs of mining.
However I think it is worth a look, if someone has nothing better to do. Just don't expect an open entrance, you would need some very heavy equipment to shift the boulders covering it. From memory, it is likely to be roughly opposite the level crossing.