My understanding is the original LPD-17 BMD concept was actually two main concepts with subvariants that did as you say, include nuclear power at times. The first main one was sort of like this with large fixed radars covering the entire horizon, and the other main concept would have mounted one or two large steering radars similar to those mounted on USNS Howard O. Lorenzen. The advantage of the later concept would have been considerably greater range and above all discrimination capability for the amount of money the radar cost via fewer but much larger antennas, but of course at the price of only sector scanning. This was attractive in the context of sea based KEI, since the odds of simultaneous threats requiring KEI engagement appearing multiple sectors would be rather low. Such a ship could stand off at very long range, at which point a 90 degree scanning sector becomes enough to cover the width of a nation the size of time.
Around the same time it was also just proposed to build a few more units of what became Howard O. Lorenzen with some level of improved capability but possibly no self defense what so ever in ordered to keep them as cheap as possible. That ship cost about 1.5 billion and an LPD-17 without BMD capability is already around 1.8 billion.