Acidic soil is not always the cause of buttercups, so you would be advised to have your soil tested before getting large amounts of lime spread by a contractor (usually at about 2 tonnes /acre)
However Calcipril or Calcifert are applied in much smaller amounts, with a conventional spreader, and will sweeten the soil without the risk of overdoing it and locking up trace elements. You can apply them at any time of year as the calcium binds to the soil particles and is not washed away by heavy rain like nitrogen is.
A suitable herbicide for buttercups is 2-4,D. It is selective for broad leaved plants, so will also kill clover and any herbs you may have as well. However, it is a plant hormone, not a poison, so will not affect your stock or compost heap.