How to compute the impulse response from incomplete frequency response data?
Phase alone or magnitude alone reconstructions of the complete frequency response (and therefore the impulse response) is claimed in IEEE papers of Monson Hayes and Alan Oppenheim such as this https://www.researchgate.net/publication/3176816_Signal_Reconstruction_from_Phase_or_Magnitude)

They are specifically dealing with real systems whose Fourier transforms have Hermitian symmetry (redundancy), therefore half of the information (phase or magnitude) in principle should be enough to define the whole system. Yet the procedure is not very simple, iterative and requires some mature DSP level.

Check it out. It may help you.