Those who say there are better options available than the three you have described are correct. However, as noted by FWJ, we often have to select the best option that is reasonably available to us. As far as I am concerned, the best option would be to HIRE SOMEONE to spray the pasture with Roundup, wait a few weeks until the weeds and grass start to grow again, spray with Roundup again, wait a few more weeks and spray a third time. Then hire someone to plant grass and clover with a no-till drill. Wait for the mixture to grow and pay someone to raise your goat and do all of the other work that needs done around your farm/home while you enjoy your family, friends and life in general. However, that isn't an option that is available to most of us.
To answer your question, I believe the disc harrow would work best. Like FWJ, I have reseeded a lot of ground with a disc harrow. I wait until the ground is moist enough to expose some soil by disturbing the ground ever so lightly. Straighten your discs out while using the disc - a little angle goes a long way. Then broadcase your seed and cultipack. It works very well and should not damage your grass; provided you do not try to do this during the heat of the summer.
You have indicated that you have Alabama clay and plan on planting white clover. Keep in mind that white clover does better in bottomland or areas that do not dry out too much during the summer. Red clover is a perennial that you might want to consider depending upon how much soil mositure you have throughout the year. The down side of red clover is that it will not likely last as many years as white clover.