We classify all compact simply connected biquotients of the form \(G/\!\!/ SU(2)^2\) for \(G =SU(4), SO(7), Spin(7)\), or \(G = \mathbf{G}_2\times SU(2)\). In particular, we show there are precisely \(2\) inhomogeneous reduced biquotients in the first and last case, and \(10\) in the middle cases.