When repairing a split table top, should I use dowels, biscuits, or something else?
Dowels are probably your best bet for this fix because the tool cost is much lower for the case that you need to purchase tools to do this repair. The cost of a doweling jig is much lower than a biscuit cutter. Drilling the dowel holes is easily accomplished with an electric drill. 

If you end up with just a one inch wide repair strip added in then I would just drill the dowel holes right through the strip and into the original material on each side. Do check your table top pieces for overall flatness after you separate the two sides from the split. Warped lumber often develops a curvature that spans a greater distance than just one inch. You'll need to take this into account when you decide to start trimming away the split edges.

With properly jointed surfaces on the two trimmed edges and the hardwood accent piece the dowels and glue should be plenty strong to hold things together.

You may want to cut the accent piece just a small bit longer than the amount needed and then trim and sand it to the existing round table edges after the gluing is done. This will give you the greatest opportunity blend the edge of the table to the nicest fit and look. 

I see some challenge in clamping the round table to get good tight glue joints. You may need to fabricate some jigs to help provide a place to engage your clamps across the width of the table. The sketch below shows one idea of how to do this. Some flat boards about 4 -> 5 inches wide can have some blocks glued and screwed on the ends that will provide a place to engage the clamps.



I have shown the blocks somewhat offset from where they would actually be attached to the boards so that you can see the nature of how they would fit to the contour of the table edge. Long bar clamps or pipe clamps would be spanned across the table top at the overhang of these boards as shown in the next picture.