As @newinvention suggests eventually I went for a compromise. The farmer and I walked round the field and we chose about a fifth of it with the smallest anthills. He took his big tractor in and flattened that area. He will come again in the autumn to mow the flattened area. It was quite traumatic walking round the field and seeing the ants trying to make sense of their flattened homes, but at least we will (hopefully) be able to manage this section for flowers.
So that just leaves the rest of it to deal with! I am following up suggestions for possible funders for stock fencing. Some require a management plan, so I need to find out what should go in one. Lots to do!
Thanks for all your interest.