I would try to have water available at all times for a lactating animal. That is their most important nutritional need.
Heavy milkers
should not be heavy fleshed, so I think you are right about her body condition being appropriate.
I raise grass-only dairy cows, but my neighbor has grass-only dairy goats and they do great. Keeping the mineral supplements available like you do is a good insurance policy.
Rotating your animals through the pasture to facilitate adequate rest periods will be really good for both the pasture and your goats. I know it seems tricky at first, but once you figure out a system, it is totally worth it. One easy way to get started is to get four lengths (whatever size seems appropriate to your situation, maybe 80') of electro-netting and a
solar fence charger. Make a paddock with two pieces, then when it is time to move the goats, set up a second paddock with the next two pieces, so that you can just open up the end of the first paddock, and the goats walk through into the second paddock. Then take down paddock number one, and set up paddock number three with that electro-net. Hope that makes sense, it is quite simple in practice and doesnt take more than a few minutes each day. This system will nullify any concerns about parasite loading in your pastures.
Interesting thing about dairy quality pastures, is that neither lush nor overgrown are ideal. It sounds like you are understocked, which is certainly better than overstocked, but still will cause problems with animal nutrition. I would reccomend reducing that area you are grazing, and rotating it with fencing. You ideally want grass that has 4 leaves and is not yet heading up. Legumes should be just starting to bloom. A good way to know for sure is to use a brix tester to know what the
energy levels are in your pasture plants. Overgrown pasture will be suprisingly low in energy, not really adequate for a lacating animal. Long and overgrown pastures can be deferred for later grazing when your animals are dry. The massive amounts of builtup biomass will then serve you well. Three good modern
books on the subject are "Management Intensive Grazing", "Grassfed to Finish", and "Quality Pasture".
hope that helps some, good luck