Magnetic bit writing by short-wave magnons without conversion to the electrical domain is expected to be a game-changer for in-memory computing architectures. Recently, the reversal of nanomagnets by propagating magnons was demonstrated. However, experiments have not yet explored different wavelengths and the nonlinear excitation regime of magnons required for computational tasks. We report on the magnetization reversal of individual 20 nm thick Ni81Fe19 (Py) nanostripes integrated onto 113 nm thick yttrium iron garnet (YIG). We suppress direct interlayer exchange coupling by an intermediate layer, such as Cu and SiO2. By exciting magnons in YIG with wavelengths λ down to 148 nm we observe the reversal of the integrated ferromagnets in a small external field of 14 mT. Magnons with a small wavelength of λ = 195 nm, i.e., twice the width of the Py nanostripes, induced the reversal at a spin-precessional power of only about 1 nW after propagating over 15 μm in YIG. Such small power value has not been reported so far. Considerations based on dynamic dipolar coupling explain the observed wavelength dependence of the magnon-induced reversal efficiency. For an increased power, the stripes reversed in an external field of only about 1 mT. Our findings are important for the practical implementation of nonvolatile storage of broadband magnon signals in YIG by means of bistable nanomagnets without the need of an appreciable global magnetic field.